11 


Southern  Branch 
of  the 

(niversity  of  California 

Los  Angeles 


Form  L 


CIA 


This  book  is  DUE  on  the  last  date  stamped  below 


OCT  2 
MAR  2  2  1931 


Form  L-9-15m-8,'26 


MATHEMATICS  FOR 

FRESHMEN  STUDENTS  OF 

ENGINEERING 


THEODORE  LINDQUIST 


o-hc  llmurraitij  nf  (thtragn 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 


MATHEMATICS 


FOR  FRESHMEN  STUDENTS 


OF  ENGINEERING 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

or  THE 
OGDEN  GRADUATE  SCHOOL  OF  SCIENCE 

IN  CANDIDACY  FOR  THE   DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

DEPARTMENT  OF  MATHEMATICS 
7 


BY 

THEODORE    UNDQUIST 


1911 


QA\\ 


CONTENTS. 

Chapter  I.    Historical  Summary 1-16 

I.     Periods  of  Engineering  History I 

II.     First  Period— to   1824 I 

1 )  The  Apprenticeship  System 1-2 

2)  The  United  States  Military  Academy 2-3 

III.  Conditions  in  Engineering  at  the  Transition  from 

the  First  to  the  Second  Period 3 

1)  Development  of  Natural  Resources 3 

2)  Transportation    

i)     Canals    3 

ii)     Railroads    4 

iii)     Steamboats  and  Bridges 4 

IV.  Second  Period— 1824  to  1862 4-5 

1 )  Purely  Technical  Institutions 5 

i)     Rensselaer  Polytechnic  Institute 5-7 

2)  Literary-Technical  Institutions    7 

i)     Union  College 7-8 

ii)     Lawrence  Scientific  School  of  'Harvard 

University    8 

iii)     Brown  University  8-9 

iv)     Cumberland  University  9 

v)     Sheffield  Scientific  School  of  Yale  Uni- 
versity   9 

vi)     University  of  Michigan 9-10 

3)  Military  and  Naval  Institutions 10 

i)     Virginia  Military  Institute IO-H 

ii)     United  States  Naval  Academy ii 

V.     Conditions  in  Engineering  at  the  Transition  from 

the  Second  to  the  Third  Period 1 1 

1 )  Transportation    12 

2)  Invention    12 


iv  MATHEMATICS  FOR  ENGINEERS. 

VI.    Third  Period— 1862  to  the  Present 12 

1)  Morrill  Land  Grant  Act  and  Subsequent  Acts  12-13 

2)  Colleges  of  the  Third  Period 13 

3)  Growth  of  the  Engineering  Colleges 13 

VII.     Sketch  of  Changes  in  Mathematics  in  Engineering 

Colleges   14 

1)  Changes  in  Entrance  Requirements 14-15 

2)  Collegiate  Curriculum 15 

3)  General  Trend   15 

VIII.     Progress  of  Instruction  in  Mathematics 16 

IX.     Summary    16 

Chapter  II.    Engineering  Colleges  in  the  United  States  in 

1908 17-49 

I.    Contents  17 

II.     Sources  of  Data 17 

III.  Explanation  of  Tables 18-20 

IV.  Tables  of  Data  on  Engineering  Colleges 21-41 

V.    Collected  Results  of  the  Tables 42 

1 )  Growth  of  Engineering  Activity 42 

2)  Entrance  Requirements  

i)     Time  for  Various  Subjects 43 

ii)     Requirements  of  Majority  of  Institutions  44 

3)  Curriculum    

i)     Position  of  Subjects 44~45 

ii)     General  Trend  of  Position  of  Subjects  in 

Curriculum    45 

iii)     Comparison  of  Time  for  Various  Sub- 
jects    46 

4)  Percentage  of  Time  Given  to  Mathematics.  .  46 

5)  Discrepancies  and  Their  Causes 46-49 

VI.     Percentage   of   Time   Given   Various   Groups   of 

Work    49 

VII.     Summary    49 

Chapter  III.    Recent  Modifications  in  the  Work  in  Mathe- 
matics for  Students  of  Engineering 50-66 

I.    Contents  of  the  Chapter 50 

II.    General  Nature  of  the  Modified  Work  in  Mathe- 
matics      50-51 


CONTENTS.  v 

III.  Sources  of  Data 51 

i )     Questionaire  Letter 52 

IV.  Synopsis  of  Principal  Features  of  Modified  Work  52 
Institution  No.     i     Coherent  Course  of  Essential 

Elements    52~53 

i)     Study  and  Use  of  Curves 53 

ii)     Early  Introduction  of  the  Calculus 53 

Institution  No.     2     Maximum  Usable  Mathemat- 
ics in  First  Year 53~54 

i)     Origin  of  Course 54 

Institution  No.     3     Mathematical  Schedule  a  Co- 
herent Whole 54 

i)     College  Algebra    54-55 

ii)     Analytic  Geometry  and  the  Calculus. ...  55 

iii)     Material  Used 55~56 

Institution  No.    4    A  Modified  Coherent  Sched- 
ule of  Subjects 56 

Institution  No.     5     Arrangement  of  Mathematics 

Courses  for  Future  Work . .  57 
Institution  No.    6    Combined  Course  in  Algebra 

and  Trigonometry  57 

i)     Mathematics  as  a  Tool 57 

Institution  No.     7    Algebra  throout  the  Course  in 

Mathematics    57 

Institution  No.    8    Early  Calculus  . 58 

Institution  No.    9     Mathematics  as  Basis  of  Tech- 
nical Training  58 

Institution  No.  10    Applied  Contrasted  with  Pure 

Mathematics    58-59 

i)     Problems  from  "Real  Material" 59 

ii)     Laboratory  Work 59 

Institution  No.  n     Nature    of    Applied    Mathe- 
matics           59-6o 

i)     Graphic  Algebra  and  Trigonometry 60 

ii)     Analytic  Geometry  and  the  Calculus. ...  60 

iii)     Problems  from  Engineering 60 

Institution  No.  12     Mathematical  Power  and  For- 
mal Course 61 

i)     Problems  from  Engineering 61 

Institutions  13  and  14    Engineering  Problems. ...  61 


vi  MATHEMATICS  FOR  ENGINEERS. 

Institution  No.  15    Extra  Problems  forEngineers  61 
Institution  No.  16    Division    into    two    Classes: 

Analysis  and  Computat'on.  61-62 

i)     Contents  of  Course  in  Computation. ...  62 

ii)     Freshman  Calculus  62 

Institution  No.  17     Problems     and     Computation 

Devices    I 62 

Institution  No.  18    Computations   62 

Institution  No.  19    Computation — Purpose    6$ 

Institution  No.  20    Laboratory  Work 63 

i)     Instruments  and  Material  Used 63 

ii)     Problems  Considered 63-64 

iii)     Working's  of  the  Course 64 

iv)     Object  of  the  Course 64 

v)     Results    64-65 

V.     Summary    65-66 

Chapter  IV.     Current  Thoughts  on  Vital  Questions 67-96 

I.     Mode  of  Obtaining  the  Data 67 

1 )  Questionaire  Letter  67-70 

2)  Replies  70 

II.     Tabulation  of  Data 70 

III.  Entrance  Examinations  71 

i)     A  Substitute  for  the  Entrance  Examination. .  71-72 

IV.  Specific  Needs  and  Deficiencies 72 

1 )  Trigonometry   

i)     Needs   73~74 

ii)     Deficiencies    74~75 

2)  Algebra    

i)     Fundamentals    75~76 

ii)     Deficiencies 7^-77 

3)  Analytic  Geometry   

i)     Object    77-78 

ii)     Importance  of  Topics 78-79 

iii)     Deficiencies    79 

iv)     Derivatives    and    Integrals    in    Analytic 

Geometry  80 

4)  Aids  and  Remedies  for  Above 80-82- 


CONTENTS.  vii 

V      Modification  in  Mathematics  for  Engineers  ....... 

i  )     Engineering  Problems  .................... 

i)     Advisability    .........................  83 

ii)     Nature  of  Such  Problems  ..............  84-85 

2)  Advisability  of  a  Course  in  Computation  .....  85-86 

3)  Segregation  of  Engineers  in  Mathematics.  .  .  . 

i)     Advisability    ......................  ...  86 

ii)     Reasons    ............................  86-87 

4)  Special  Texts  ............................  87 

VI.     Pedagogic  Questions   .......................... 

1)  Quizzes   .................................  88 

2)  Modes  of  Conducting  Classes  ..............  88-90 

VII.     General  Needs  and  Deficiencies  .................  90 

i  )     General  Mathematical  Ability  ............... 

i)     Need    ..............................  90-91 

ii)     Deficiencies    .........................  92 

iii)     Remedies    ...........................  92-94 

2)     Relative  Importance  of  Topics  ..............  94 

VIII.     Engineering  vs.  a  General  Education  ............  94~95 

IX.     Summary   .................................... 


Chapter  V.    Present  Needs  and  Tendencies  .............  97-121 

I.     Contents  of  Chapter  ..........................  97 

II.     Preparatory  Mathematics  ......................  97 

i  )     Nature  of  Work  Needed  ...................  97-9$ 

i)     Algebra  to  be  Treated  as  a  Study  of 

Equations    ........................  9&~99 

2)  Deficiencies  Found  .......................  99 

3)  Cause  of  Deficiencies  .....................  99 

i)     School  Management  .................  100 

ii)     The  Teacher  ........................  100 

4)  Suggested  Remedies  ......................  100-101 

i)     Requirement   of   Entrance   Examination 

in  Mathematics  ....................  101 

ii)     Substitute  for  Entrance  Examinations.  .  101 

iii)     Demands  upon  the  High  Schools  .......  101-102 


viii  MATHEMATICS  FOR  ENGINEERS. 

III.    Collegiate  Mathematics  102 

1)  What  Should  be  Taught 102 

2)  How  It  Should  be  Taught 102 

3)  Specific  and  General  Needs  and  Difficulties. .  102-103 

4)  Specific  Needs  and  Difficulties 103 

i)     Trigonometry 

a)  Reason  for  Place 'in  Curriculum...  103 

b)  Needs    103-104 

c)  Deficiencies    104 

d)  Suggestions  for  Improvement 104 

e)  Trigonometry   to  be   Taught   from 

the  Utilitarian  Standpoint 104-105 

f )  Suggested  Course 105 

ii)     Algebra 

a)  Status  in  Engineering  Colleges. . . .  105 

b)  Needs  and  Difficulties  106 

c)  Suggested  Improvements  

I)     Teach  Elements 106 

II)     Principles  of  Algebra  to  be 

Taken  up  When  Needed. .  107 

III)     Algebra  for  Future  Use....  107-108 

iii)     Analytic  Geometry   

a)  As  a  Mathematical  Language 108-109 

b)  Illustrative  Problems  109 

c)  Other  Devices   1 10-1 1 1 

d)  Needs  and  Difficulties in 

iv)     The  Calculus   

a)  Principles  Taught 111-112 

b)  Advantages   112 

v)     Combined  Course  in  Algebra,  Coordinate 

Geometry  and  Elements  of  the  Calcu- 
lus    i 12-1 13 

vi)     Work  in  Computation 

a)  Mode  of  Presentation 113-114 

b)  Degrees  of  Accuracy 114-115 

5)  General  Needs  and  Difficulties 115 

i)     Cooperation  with  the  Professional  De- 
partments      115-116 


CONTENTS.  ix 

ii)     Problems  

a)  Their  Use 1 16 

b)  Kinds  of  Problems 116-117 

c)  Problems  to  Develop  Mathematical 

Thought    117 

6)     Pedagogic  Considerations  117 

i)     Modes  of  Instruction 117-118 

ii)     Suggested  Combinations  of  Modes 118-119 

iii)     Special  Texts  in  Mathematics 119-120 

iv)     The  Profession  of  Teaching 120 

V.     Summary    121 

Chapter  VI.    Conclusions  122-130 

I.     Contents  of  Chapter 122 

II.     Origin  and  Growth  of  the  Engineering  Colleges. .  122 
III.     Progress  of  Work   in   Mathematics   and   Present 

Needs  123 

1 )  Entrance  Requirements  

i)     Their  Scope   123 

ii)     Present  Needs   123-124 

iii)     Remedy   124 

iv)     Usefulness  of  Variation  in  Entrance  Re- 
quirements      124-125 

2)  Collegiate  Curriculum 

i)     Progress    125 

ii)     Suggested  Improvements 126 

3)  Instruction    

i)     Progress  and  Present  Needs 126-127 

ii)     Special  Phases  to  be  Noted 127 

a)  General  Aims  127 

b)  Fundamental   Principles    127 

c)  Problems    127-128 

d)  Computation   128-129 

4)  Too  Much  Required  of  the  Mathematical  De- 

partment     129 

III.     Mathematics  for  Freshmen  Students  of  Engineer- 
ing:    129-130 

Bibliography    131-135 


MATHEMATICS  FOR  FRESHMEN 
STUDENTS  OF  ENGINEERING 


-2,  £  4^<£  7 
CHAPTER  I. 


HISTORICAL  SUMMARY. 

I.    PERIODS  OF  ENGINEERING  HISTORY. 

The  history  of  engineering  education  in  the  United  States  falls 
into  three  periods  separated  by  two  notable  events.  The  first  of 
these  is  the  establishment,  in  the  first  part  of  the  nineteenth  century, 
of  courses  in  civil  as  distinct  from  military  engineering ;  the  second 
the  passage  of  the  Land  Grant  Act  in  1862.  While  in  this  case,  as 
in  many  others,  succeeding  periods  blend  into  each  other  without 
any  sharp  line  of  demarkation,  yet  these  three  periods  are  very  real 
and  distinct. 

II.    FIRST  PERIOD— TO  1824. 

f 

During  the  first  period  there  were  two  sources  of  engineering 
education ;  the  apprenticeship  system  and  the  United  States  Military 
Academy. 

i)  THE  APPRENTICESHIP  SYSTEM. 

The  apprenticeship  system  was  confined  chiefly  to  New  Eng- 
land; it  continued  in  existence  until  the  Civil  War  and  furnished 
most  of  the  engineering  education  up  to  about  1835. 


2  MATHEMATICS  FOR  ENGINEERS. 

Instruction  consisted  of  such  information  as  the  apprentice 
could  "pick  up"  through  questions  in  connection  with  field  and  office 
work,  for  which  privilege  he  paid  one  hundred  dollars  per  year  for 
a  term  of  three  years.  For  his  work  in  the  field  he  received  twelve 
and  one-half  cents  per  hour  as  compensation,  but  before  the  begin- 
ning of  the  Civil  War  nothing  was  paid,  as  a  rule,  for  office  work. 
After  that  date  all  work  was  paid  for  at  the  rate  mentioned.  The 
above  conditions  were  quite  uniformly  prevalent.  This  system 
presented  the  extreme  of  "practical"  instruction  to  the  exclusion 
of  all  theory. 

2)  THE  UNITED  STATES  MILITARY  ACADEMY. 

The  United  States  Military  Academy  was  one  of  the  main 
sources  of  engineering  education  during  this  period  since  many  of 
its  graduates  took  up  work  along  civil  lines,  and  remained  a  factor  in 
engineering  education  until  1850. 

As  early  as  September,  1776,  a  committee  of  the  Continental 
Congress  recommended  a  bill  for  a  "Continental  Laboratory  and 
Military  Academy."  Later  Washington,  as  president,  several  times 
urged  Congress  to  establish  such  an  institution.  Nothing  was  done, 
however,  until  1802  when  the  bill  authorizing  its  establishment  was 
passed,  March  16.  July  4,  of  the  same  year,  the  Academy  was 
opened  at  West  Point  with  ten  cadets. 

At  first  the  instruction  in  mathematics  was  very  limited  extend- 
ing only  through  Hutton's  first  volume.  By  1810  instruction  included 
arithmetic,  logarithms,  elementary  algebra,  geometry,  trigonometry, 
mensuration  of  heights  and  distances,  planimetry,  stereotomy,  sur- 
veying and  conic  sections.  In  1816  a  four  years'  schedule  of  studies 
was  adopted  in  which  the  mathematical  work  "embraced  the  follow- 
ing branches,  namely:  the  nature  and  construction  of  logarithms, 
and  the  use  of  the  tables;  algebra,  to  include  the  solution  of  the 
cubic  equation,  with  all  preceding  rules ;  geometry,  to  include  plane 
and  solid  geometry,  also  ratio  and  proportion,  construction  of 
geometrical  problems,  application  of  algebra  to  geometry,  practical 
geometry  on  the  ground,  mensuration  of  the  planes  and  solids ;  plane 
trigonometry,  with  the  application  to  surveying,  and  the  mensuration 
of  heights  and  distances ;  spherical  trigonometry,  with  its  application 
to  spherical  problems :  the  doctrine  of  infinite  series ;  conic  sections 


HISTORICAL  SUMMARY.  3 

with  their  application  to  projectiles;  fluxions  to  be  taught  at  the 
option  of  the  professor  and  students."  "Fluxions,"  however,  were 
seldom  given.  This  curriculum  also  included  "A  complete  course 
in  engineering."1  From  1816  there  has  been  a  gradual  growth  in  the 
amount  of  mathematics  offered,  up  to  the  present  schedule,  which 
was  put  into  operation  with  the  class  entering  March,  icjoS.2 

III.    CONDITIONS  IN  ENGINEERING  AT  THE  TRANSI- 
TION FROM  THE  FIRST  TO  THE 
SECOND  PERIOD. 

A  brief  glance  at  conditions  in  the  engineering  world  about  the 
time  of  transition  from  the  first  to  the  second  period  may  be  of 
interest. 

1)  DEVELOPMENT  OF  NATURAL  RESOURCES. 

The  output  of  iron  had  risen  from  54,000  tons  in  1810  to 
165,000  tons  in  1830  and  to  347,000  tons  in  1840.  Coal  mining  was 
begun  about  1820  with  an  output  of  65,000  tons  for  that  year.  Pro- 
gress in  agricultural  developments  gave  rise  to  the  manufacture  of 
farm  implements;  plows  and  mowers,  of  quite  good  pattern,  were 
produced  as  early  as  1820. 

2)  TRANSPORTATION — (i)  Canals. 

The  first  attempt  at  a  solution  of  the  transportation  problem  in 
this  country  was  the  building  of  canals.  As  early  as  1793  a  short 
canal  was  constructed  in  order  to  make  a  contour  around  the  falls 
in  the  Connecticut  river  at  South  Hadley,  Massachusetts.  The  first 
freight  and  passenger  canal,  which  was  three  miles  in  length,  was 
begun  the  same  year  and  finished  in  1804,  but  has  not  been  in  use 
since  1850.  The  principal  canal  building  era  lasted  from  1810  to 
about  1840.  Only  one  canal  of  importance,  the  Illinois  and  Mich- 
igan, in  course  of  construction  from  1836  to  1848  was  finished  after 
this  time. 

1  Bvt.  Maj.-Gen.  G.  W.  Cullum,  Biographical  Register  of  the  Officers  and  Gradu- 
ates of  the  U.  S.  Military  Academy  with  the  Early  History  of  the  United  States  Mili- 
tary Academy.  Boston  and  New  York,  1891. 

*  Chap.  II,  p.  41. 


4  MATHEMATICS  FOR  ENGINEERS. 

(«)  Railroads. 

Canal  building  decreased  as  a  result  of  the  coming  of  the  rail- 
road. The  first  railroad,  three  miles  in  length,  was  constructed  in 
1826  at  a  cost  of  $34,000  and  used  horses  as  motive  power.  In  1829 
the  first  English  made  locomotive  was  imported  and  used  on  a  six- 
teen mile  track.  A  year  later  the  first  American  locomotive  was 
constructed.  The  second  American  locomotive  was  -used  by  the 
South  Carolina  Railroad  which  ran  between  Charleston  and  Ham- 
burg. On  its  first  passenger  trip,  made  January  15,  1831,  it  attained 
a  speed  of  15  to  20  miles  per  hour.  The  following  table  gives  the 
mileage  of  railroads  up  to  1900: 

1830    23  miles 

1831     95  miles 

1832    229  miles 

1835     1,098  miles 

1840    2,818  miles 

1845  4,633  miles 

1850  9,021  miles 

1860  30,626  miles 

1870  52,922  miles 

1900  190,082  miles 

(iii)  Steamboats  and  Bridges. 

The  application  of  steam  to  water  traffic  was  made  by  Robert 
Fulton  with  his  Clermont,  as  early  as  1807.  It  was  not  used  for 
transatlantic  ships,  however,  before  1839.  The  building  of  tunnels 
and  bridges  which  had  just  begun  at  this  time  also  greatly  faciliated 
traffic. 

IV.    SECOND  PERIOD— 1824  TO  1862. 

During  the  first  years  of  the  nineteenth  century  a  great  step 
forward  was  made  in  education  in  the  United  States  by  the 
introduction  of  laboratory  instruction.  The  principle  involved  was 
further  developed  in  the  organization  of  courses  giving  work  in 
civil  engineering.  The  Rensselaer  School,  founded  by  Stephen  Van 
Rensselaer  in  1824,  was  the  pioneer  in  this  movement.  The  date  of 
its  founding,  therefore,  serves  to  indicate  the  beginning  of  the  second 


HISTORICAL  SUMMARY.  5 

period,  although  this  period  was  not  fully  established  until  some 
years  later.  Following  the  establishment  of  the  Rensselaer  School 
several  of  the  literary  institutions  then  existing  organized  courses 
in  engineering.  As  a  consequence  there  are  to  be  found  two  distinct 
classes  of  technical  institutions  which  will  here  be  considered  briefly ; 
purely  technical  institutions  and  literary-technical  institutions. 

i)  PURELY  TECHNICAL,  INSTITUTIONS. 

(t)  Rensselaer  Polytechnic  Institute. 

The  Rensselaer  School  was  the  only  purely  technical  institution 
founded  during  this  period.  Concerning  its  object,  Stephen  Van 
Rensselaer  wrote  Rev.  Samuel  Blatchford  of  Lansingburg,  October 
5,  1824,  that  he  had  established  a  school  for  those  "who  may  choose 
to  apply  themselves  in  application  of  science  to  the  common  purposes 
of  life.  My  object  is  to  qualify  teachers  for  instructing  the  sons 
and  daughters  of  farmers  and  mechanics,  by  lectures  and  otherwise 
in  the  application  of  experimental  chemistry,  philosophy  and  natural 
history  to  agriculture,  domestic  economy,  the  arts  and  manufacture." 
The  institution  has,  however,  devoted  itself  to  the  direct  preparation 
of  its  students  for  technical  occupations.  The  original  name,  Rens- 
selaer School,  was  changed  in  1833  to  Rensselaer  Institute.  Instruc- 
tion in  civil  engineering  was  first  offered  in  the  annual  announce- 
ment for  1828  but  no  special  course  was  established  before  1835. 
The  institution  was  again  reorganized  in  1851,  when  the  present 
name,  Rensselaer  Polytechnic  Institute,  was  adopted.  At  this  time 
the  course  of  study  was  also  extended  from  one  to  three  years.  In 
1852  there  was  added  a  preparatory  course  of  one  year,  which  was 
abolished  in  1862  when  the  course  was  lengthened  to  four  years. 

In  the  mode  of  instruction  the  Rensselaer  School  made  a  very 
decided  departure  from  that  of  the  other  institutions  existing  at  the 
time  of  its  organization.  The  mode  was  in  substance  as  follows: 
each  new  topic  was  taken  up  by  the  professor  in  charge  in  a  lecture 
upon  which  the  class  was  quizzed  the  following  day,  the  class  was 
then  divided  into  sections  which  met  in  various  smaller  rooms.  At 
the  meetings  of  each  of  these  sections  a  student  would  be  selected  to 
repeat  the  kcture  of  the  day  previous.  Although  an  increased 
number  of  students  has  compelled  a  change  in  the  form  of  carrying 
on  the  work  to  that  of  the  lecture  and  quiz,  still  the  spirit  of  the  orig- 


6  MATHEMATICS  FOR  ENGINEERS. 

inal  mode  has  been  retained.  This  spirit — that  of  "Learning  by 
doing" — is  the  feature  to  be  emphasized  in  connection  with  this  insti- 
tution.8 

The  following  is  an  outline  of  the  course  of  study  offered  in 
1854  after  the  course  in  civil  engineering  had  fully  crystalized.  It 
is  to  be  regretted  that  the  records  do  not  show  the  number  of  exer- 
cises devoted  to  each  subject. 

First  Year. 

First  Term — Algebra,  geometry,  general  physics,  graphics,  geo- 
desy, English  composition,  French. 

Second  Term — Trigonometry,  higher  algebra,  general  chemis- 
try, graphics,  geodesy,  botany,  English  composition, 
French. 

Second  Year. 

First  Term — Analytics,  differential  calculus,  practical  trigonom- 
etry, general  physics,  minerology,  chemistry,  descriptive 
geometry,  English  composition,  French. 

Second  Term — Integral  calculus,  general  physics,  geology,  zool- 
ogy,  graphics,  geodesy,  English  composition,  German. 

Third  Year. 

First  Term — Mechanics,  astronomy,  physical  geography,  geol- 
ogy, industrial  physics,  English  composition,  philosophy. 
Second  Term — Construction,  mechanics,  mining,  geodesy,  prac- 
tical astronomy,  graphics,  metallurgy,  industrial  physics, 
philosophy. 

At  first  those  "who  have  a  good  knowledge  of  arithmetic  and 
can  understand  good  authors  readily"  were  received  into  the  institu~ 
tion.  The  requirements  were  raised,  however,  until  at  the  time  the 
above  schedule  of  studies  was  put  into  operation  they  were,  in  math- 
ematics, arithmetic,  including  the  metric  system,  plane  geometry  and 
algebra  to  equations  of  the  second  degree.  In  view  of  the  fact  that 
a  preparatory  course  of  one  year  was  offered  at  this  time  and  that 
the  college  course  was  only  three  years  in  length  these  were  really 
higher  than  the  actual  requirements  for  entrance  to  the  institution. 
Davies'  Legendre's  Geometry,  Davies'  Bourdon's  Algebra,  Chauv- 

•  Palmer  C.   Rickets,  History  of  the  Rensselaer  Polytechnic  Institute   1824-94.     New 
York,  1895. 


HISTORICAL  SUMMARY.  7 

enet's  Trigonometry,  Church's  Analytical  Geometry  and  Church's 
Calculus  were  the  texts  used.  By  1893  the  work  in  mathematics 
had  been  changed  to  solid  geometry,  trigonometry  and  algebra  for 
the  first  year,  analytical  geometry  for  the  second,  and  the  calculus 
for  the  third.  Two  years  later  the  differential  calculus  was  put  into 
the  second  year.4 

2)  LITERARY-TECHNICAL  INSTITUTIONS. 

About  the  middle  of  the  nineteenth  century  several  literary 
institutions  recognized  the  advisability  of  adding  courses  in  engi- 
neering. As  a  rule  these  were  at  first  a  modification  of  the  final 
two  years  of  their  existing  courses,  but  they  gradually  developed 
into  the  highly  specialized  ones  of  the  present.  Because  of  the  sim- 
ilarity of  development,  changes  in  the  schedule  will  be  given  fully 
for  two  institutions  and  only  special  characteristics  of  the  others. 

(t)  Union  College. 

Union  College,  at  Schenectady,  N.  Y.,  was  established  as  early 
as  1795,  but  the  Department  of  Civil  Engineering  was  not  organized 
before  1845.  Until  1852  the  work  required  in  mathematics  was 
algebra  (Bourdon),  one  year;  plane  geometry  (Legendre),  one- 
third  year ;  solid  geometry,  one-third  year ;  plane  and  spherical  trig- 
onometry, one-third  year.  Beginning  with  1852  one-third  year  was 
added  in  each  of  the  following:  algebra,  conies  (Jackson),  analytic 
geometry  of  three  dimensions  (Davies)  and  the  calculus  (Davies). 
A  little  later  the  course  was  changed  so  as  to  call  for  only  two  years, 
but  the  entrance  requirements  were  raised  so  as  to  include  a  large 
amount  of  work  previously  given  in  the  four  years'  course.  The 
entrance  requirements  in  mathematics  were  then  two  terms  each  of 
algebra  and  geometry,  and  one  term  each  of  plane  and  spheri- 
cal trigonometry  and  geometrical  drawing.  During  the  first 
year  of  the  two  year  course  one  term  was  given  to  each  of  the  fol- 
lowing: analytic  geometry,  accurate  and  approximate  calculations 
and  the  calculus.  In  1875  the  course  was  again  lengthened  to  four 
years  and  the  entrance  requirements  lessened,  so  that  in  mathematics 
algebra,  to  equations  of  the  second  degree,  was  the  only  requirement. 
In  1890  this  was  increased  to  algebra  thru  equations  of  the  second 

4  The   data   for   this   institution   and   for   the   others   to   be   considered   are   found   in 
past  catalogs  and  announcements. 


S  MATHEMATICS  FOR  ENGINEERS. 

degree  and  plane  geometry.  Since  that  time  various  changes  in  the 
entrance  requirements  in  mathematics  have  been  made  until  now 
they  stand  as  given  in  Chapter  II  where  will  also  be  found  the  work 
in  mathematics  as  scheduled  for  the  college  courses.5 

(it)  Lawrence  Scientific  School  of  Harvard  University. 

In  1847  Abbott  Lawrence  gave  $50,000  for  the  founding  of  a 
scientific  school  in  connection  with  Harvard  University.  He  desired 
"a  school  for  the  purpose  of  teaching  the  practical  sciences — I  have 
thought  that  these  great  branches  to  which  a  scientific  education  is 
to  be  applied  amongst  us  should  be  first,  engineering ;  second,  mining 
in  its  extended  sense,  including  metallurgy ;  third,  the  invention  and 
manufacture  of  machinery."  In  honor  of  the  donor  this  newly 
organized  school  was  called  the  Lawrence  Scientific  School.  The 
course  in  civil  engineering  was  not  begun  until  two  years  after  the 
founding  of  the  school.  The  course  in  mining  was  first  offered  in 
1868,  electrical  engineering  in  1888,  mechanical  engineering  in  1894, 
and  architecture  in  1895.  In  1907  the  courses  in  the  Lawrence  Sci- 
entific School  were  made  a  part  of  the  courses  offered  by  Harvard 
College  and  Graduate  School,  thus  making  the  course  in  engineer- 
ing largely  elective. 

(m)  Brown  University. 

Brown  University  offered  its  first  work  in  engineering  in  1849  in 
the  form  of  a  partial  course  of  two  years  called  the  "Engineering 
and  Scientific  Course."  The  work  in  mathematics  included  algebra, 
plane  and  solid  geometry,  mensuration,  trigonometry,  surveying  and 
mechanics.  The  following  year  a  subscription  of  $25,000  was  raised 
and  the  University  reorganized  by  Pres.  Wayland,  at  which  time  the 
"New  System"  was  adopted,  which  was  in  substance  our  full  elective 
plan.  One  and  one-half  years'  professional  work  in  civil  engineering 
was  then  offered  which  required  as  preparation  about  two  full  years 
of  mathematics.  At  the  same  time  a  course  in  practical  "chemistry 
applied  to  the  arts"  was  also  organized.  In  1863  Brown  University 
was  made  one  of  the  beneficiaries  of  the  government  in  accordance 
with  the  land  grant  act  of  1862.  Singularly  enough  no  reference 
to  work  in  engineering  is  found  in  the  catalogs  from  1863  to  1867. 
The  catalog  for  1867  offered  a  mechanical  and  agricultural  course 

•  Chap.  II,  pp.  21-41. 


HISTORICAL  SUMMARY.  9 

which  gave  in  the  mechanical  division  geometry  and  algebra,  the 
first  year;  trigonometry  and  engineering,  the  second  year;  besides 
several  culture  studies  thruout  the  course.  The  final  change  in  the 
mathematical  schedule  was  made  in  1901. 

(iv}  Cumberland  University. 

In  1852  Cumberland  University  added  a  School  of  Engineering 
to  its  other  departments. 

(v}  Sheffield  Scientific  School  of  Yale  University. 

The  Scientific  School  of  Yale  University  dates  from  the  estab- 
lishment of  its  School  of  Applied  Chemistry  in  1847.  The  "Course 
in  Engineering"  of  two  years  was  first  offered  in  1853,  and  in  1864 
was  lengthened  to  three  years  with  a  variation  along  the  lines  of  civil 
and  mechanical  engineering  during  the  last  two  years.  Mining  engi- 
neering was  added  in  1866,  electrical  engineering  in  1894  and  sani- 
tary engineering  in  1900.  The  usefulness  of  the  school  was  greatly 
enhanced  by  a  liberal  endowment  in  1860  from  Joseph  E.  Sheffield, 
in  whose  honor  it  took,  two  years  later,  the  name  of  the  Sheffield 
Scientific  School.  In  1864  it  received  congressional  recognition  and 
was  made  a  participant  in  the  aid  given  the  "Land  Grant  Colleges." 
This  revenue  was  taken  from  it  by  the  Connecticut  Legislature  in 
1892. 

The  chief  feature  in  the  mathematical  schedule  was  the  intro- 
duction in  1864  of  the  calculus  of  variations  and  in  1874  of  "Elemen- 
tary Theory  of  Numerical  Approximations,  Solutions  of  Higher 
Numerical  Equations,  Methods  of  Interpretations"  for  the  second 
term  of  the  first  year.  In  1880  the  schedule  in  mathematics  stood 
as  follows :  first  year,  analytic  geometry  and  spherical  trigonometry, 
one  semester  each;  second  year,  elementary  theory  of  functions, 
numerical  equations  and  differential  and  integral  calculus,  one  semes- 
ter each.  The  following  year  geometry  of  three  dimensions  was 
put  into  the  second  year.  In  1902  the  first  year  was  changed  further 
to  plane  and  solid  analytic  geometry,  or  an  introduction  to  the  cal- 
culus. 

(rt)   University  of  Michigan. 

The  University  of  Michigan  holds  the  honor  of  being  the  first 
institution  supported  by  a  state  to  give  an  engineering  course.  It 
was  also  the  only  one  of  its  kind  before  the  passage  of  the  Land 


I0  MATHEMATICS  FOR  ENGINEERS. 

Grant  Act  in  1862.  In  1853  its  first  course  in  civil  engineering 
was  offered  in  connection  with  the  general  science  course.  The 
work  in  mathematics  was  algebra,  two  terms,  geometry,  two 
terms,  during  the  first  year;  trigonometry,  one  term,  conies, 
two  terms  the  second  year;  the  calculus,  one  term  the  third 
year.  During  1856-57  the  two  semester  plan  was  put  into  effect 
and  the  mathematics  schedule  contained  algebra  and  geometry  the 
first  half  and  geometry,  trigonometry  and  mensuration  the  second 
half  of  the  first  year;  descriptive  and  analytic  geometry  the  first 
half  and  the  calculus  the  second  half  of  the  second  year.  In  the 
catalog  for  the  year  1859  this  statement  is  found:  "The  School  of 
Engineering  commences  with  the  second  year  of  the  Scientific  Course 
and  is  identical  with  it  during  the  second  and  third  years  of  that 
course."  The  additional  years  of  technical  work  were  then  added 
which  made  algebra  and  geometry  required  subjects  and  placed  trig- 
onometry and  analytic  geometry  in  the  first  year  of  the  engineering 
course,  with  the  calculus  in  the  second  year,  each  of  which  was  pur- 
sued for  one-half  year.  A  year  later  trigonometry  was  placed  in  the 
first  year  of  the  scientific  course  and  hence  required  for  entrance  to 
the  engineering  course.  The  School  of  Mines  was  established  in 
1865  and  made  a  separate  institution  in  1885.  In  1865  the  first  three 
years  of  the  Engineering  and  Scientific  Courses  were  also  made  iden- 
tical. This  brought  geometry  (5-books),  trigonometry  and  algebra 
to  the  first,  analytic  geometry  and  the  calculus  to  the  second  year  of 
the  engineering  course.  Euclidean  geometry  was  required  for 
entrance  after  1866,  and  plane  trigonometry  after  1890.  In  1896  the 
School  of  Engineering  was  made  a  wholly  separate  department. 

3)  MILITARY  AND  NAVAL  INSTITUTIONS. 

For  the  sake  of  completeness  two  institutions  established  during 
this  period,  the  Virginia  Military  Institute  and  the  United  States 
Naval  Academy,  although  not  organized  to  give  work  in  civil  engi- 
neering will  be  considered  briefly. 

(i)   Virginia  Military  Institute. 

Virginia  Military  Institute,  which  was  established  as  early  as 
1839,  was  patterned  largely  after  the  United  States  Military  Acad- 
emy. In  1860,  thru  bequests,  it  expanded  along  lines  of  general 
industrial  education,  only  to  have  its  plant  destroyed  by  the  northern 


HISTORICAL  SUMMARY.  u 

army.  The  cadets  were  transferred  to  Richmond,  where  the  work 
was  continued  until  the  evacuation  of  that  city  in  1865.  Since  the 
reopening  of  the  Institute  in  October,  1865,  at  Lexington,  it  has  from 
time  to  time  been  enlarging  its  powers.  The  present  courses  for 
the  first  two  and  one-half  years  are  the  same,  after  which  the  student 
elects  one  of  the  following  courses ;  chemical,  electrical  or  civil  engi- 
neering. 

(«')   United  States  Naval  Academy. 

The  establishment  of  a  Naval  Academy  was  first  suggested  by 
Hon.  William  Jones,  Secretary  of  the  Navy  as  early  as  1814,  but 
it  was  not  until  1845  that  the  present  school  was  created  at  Annapo- 
lis. Preceding  this  time  the  Navy  had  conducted  a  sort  of  midship- 
man apprentice  course.  As  in  all  such  courses,  the  instruction  was 
largely  practical,  all  of  the  theoretical  matter  being  taken  during  the 
last  six  months  of  the  course.  The  work  prescribed  in  mathematics 
for  this  brief  period  was  books  I,  2,  3,  4,  and  6  of  Ray  fair's  Euclid, 
McClure's  Spherics  and  Bourdon's  Algebra.  The  Department  of 
Mathematics  in  the  new  school  was  fully  organized  by  1850  and  gave 
work  in  arithmetic  and  algebra  the  first  year;  geometry,  trigonom- 
etry and  descriptive  geometry  the  second  year;  analytic  geometry, 
the  calculus  and  astronomy  the  third  year;  navigation  and  survey- 
ing the  fourth  year.  From  1866  to  1870  little  was  done  with  ana- 
lytic geometry.  Arithmetic  was  omitted  in  1871,  at  which  time  the 
calculus  and  mechanics  were  organized  into  one  course.6 


V.    CONDITIONS  IN  ENGINEERING  AT  THE  TRANSI- 
TION FROM  THE  SECOND  TO  THE 
THIRD  PERIOD. 

Toward  the  close  of  the  second  period  conditions  in  engineering 
were  changing  at  a  rapid  rate  corresponding  to  external  influences. 
By  1860  the  consumption  of  iron  had  arisen  to  919,370  tons.  The 
growth  and  general  development  of  the  country  was  marvelous 
which  resulted  in  many  timesaving  inventions. 

«  Park  Benjamin,  United  States  Naval  Academy.  New  York,  1900.  J.  R.  Soley,  Rear- 
Admiral  U.  S.  N.,  Historical  Sketch  of  the  United  States  Naval  Academy.  Government 
Printing  Office,  1876. 


12  MATHEMATICS  FOR  ENGINEERS. 

1)  TRANSPORTATION. 

The  first  iron  truss  bridge  is  thought  to  have  been  built  by 
Trumbull  in  1840  over  the  Erie  Canal.  During  the  ten  years  from 
1850  to  1860  the  railroad  mileage  had  increased  about  250%. 7  Two 
inventions  closely  related  to  the  railroad  industry  were  made  at  this 
time ;  the  telegraph  in  1844  and  the  air-brake  in  1865. 

2)  INVENTIONS. 

The  threshing  machine  was  perfected  about  1850  and  the  reaper 
ten  years  later.  A  sucessful  fire  engine  was  made  in  1853.  Great 
advances  were  made  in  dynamo  machinery  from  1860  to  1870.  The 
two  most  notable  events  were  the  invention  of  cassions  by  M.  Triger 
just  before  the  middle  of  the  century  and  the  establishment  of  a  plant 
for  the  manufacture  of  steel  by  the  Bessemer  process  in  1859. 


VI.    THIRD  PERIOD— 1862  TO  THE  PRESENT. 

The  third  period  is  that  of  the  land  grant  colleges.  They  are 
the  result  of  the  "Morrill  Land  Grant  Act"  passed  by  Congress  in 
1862  in  recognition  of  the  need  of  technical  education  thruout  the 
United  States. 

i)  MORRILL  LAND  GRANT  ACT  AND  SUBSEQUENT  ACTS. 

In  substance  the  Morrill  Land  Grant  Act  is  as  follows :  A  grant 
of  land  was  to  be  made  to  each  state  in  the  Union  in  the  amount  of 
30,000  acres,  or  its  equivalent,  for  each  senator  and  representative 
in  Congress  to  which  the  state  was  entitled  by  the  apportionment  of 
the  census  of  1860.  The  proceeds  from  the  sale  of  these  lands  for 
each  state  were  to  form  an  endowment  for  the  institutions  estab- 
lished under  the  provisions  of  the  act,  and  only  the  interest  derived 
from  the  same  to  be  available  for  the  support  of  these  institutions. 
It  further  required  of  such  colleges  that  their  "leading  objects  shall 
be,  without  excluding  other  scientific  and  classical  studies,  and 
including  military  tactics,  to  teach  such  branches  of  learning  as  are 
related  to  agriculture  and  the  mechanical  arts,  in  such  a  manner  as 

1  See  Table  p.  4. 


HISTORICAL  SUMMARY.  !3 

the  legislatures  of  the  states  may  respectively  prescribe,  in  order  to 
promote  the  liberal  and  practical  education  of  the  industrial  classes 
in  the  several  pursuits  and  professions  in  life."  In  1892  Congress 
passed  another  bill  known  as  the  "Morrill  Fund  Act:"  "to  apply  a 
portion  of  the  proceeds  of  the  public  lands  to  the  more  complete 
endowment  and  support  of  the  colleges  for  the  benefit  of  agriculture 
and  the  mechanics  arts  established  under  the  provisions  of  the  act 
of  1862."  According  to  this  act  the  sum  of  $15,000  was  to  be  paid 
each  of  the  land  grant  colleges  in  1890,  and  further,  this  was  to  be 
increased  by  $1,000  each  year  until  1900,  after  which  the  sum  of 
$25,000  should  be  appropriated  annually.  March  4,  1907  Congress 
again  increased  this  amount  by  what  is  known  as  the  "Nelson 
Amendment."  It  provided  that  beginning  July  i,  1908,  the  sum  of 
$5,000  was  to  be  added  to  each  appropriation  yearly  until  the  amount 
was  $5o,ooo.8 

2)  COLLEGES  OF  THE  THIRD  PERioo.9 

No  review  will  be  made  of  the  colleges  established  during  the 
third  period,  for  their  progress  presents  no  radical  changes,  except 
at  a  very  late  date  to  which  Chapter  III  will  be  devoted.  A  sum- 
mary of  the  work  offered  by  these  various  institutions  will  be  found 
in  the  next  chapter.10 

3)  GROWTH  OF  THE  ENGINEERING  COLLEGES. 

The  graph  giving  the  increase  in  the  number  of  engineering 
colleges  serves  best  to  show  the  marvelous  growth  in  engineering 
education  since  the  recognition  of  its  need  by  the  United  States 
Government.11  The  following  figures  regarding  the  increase  in 
students  taking  professional  courses  from  1878  to  1900  may  be  of 
interest : 

theology  ' 8,079  or    %7% 

medicine     26,088  or  142% 

law    1 1,835  or  294% 

engineering    9,659  or 


•  I.  E.  Clark,  A.M.,  Education  in  the  Industrial  and  Fine  Arts  in  the  United  States, 
Vol.  IV,  p.  838.     U.  S.  Bureau  of  Education,  1898. 

•  For  list  of  Land  Grant  Colleges  see  foot-note  p.  21. 
10  Chap.  II,  pp.  21-41. 

»  Chap.  II,  p.  42. 

12 1.   O.   Baker,   Presidential  Address  at  the  Annual  Meeting  of  the  Society  for  the 
Promotion  of  Engineering  Education,  1900.     Proceedings  vol.  VIII,  p.   n. 


1 4  MATHEMATICS  FOR  ENGINEERS. 

VII.    SKETCH  OF  CHANGES  IN  MATHEMATICS  IN 
ENGINEERING  COLLEGES. 

The  engineering  courses  offered  by  all  of  the  institutions  exist- 
ing during  the  second  period,  with  the  exception  of  Rensselaer  Poly- 
technic Institute,  were  only  continuations  of  their  previously  existing 
literary  or  general  science  courses  in  which  engineering  subjects  had 
been  introduced  into  the  work  of  the  last  two,  or  at  most,  three 
years.  For  this  reason  the  work  in  mathematics  for  engineers  really 
began  with  that  given  in  the  general  science  course  in  those  institu- 
tions where  the  engineering  course  was  a  continuation  of  the  former. 

The  second  period  may  be  regarded  as  the  formative  and  expe- 
rimental stage  in  the  development  of  the  engineering  colleges; 
changes  in  the  mathematical  work  were  numerous, '  sometimes 
increasing  often  diminishing  the  amount  required  for  entrance  with 
corresponding  shifting  of  the  collegiate  curriculum.  During  the 
third  period  changes  in  entrance  requirements  have  always  been  in 
the  nature  of  increases.  During  this  third  period  there  have  arisen 
institutions  with  widely  differing  purposes,  presenting  a  wide  range 
of  variation  both  in  the  work  required  for  entrance  as  well  as  that 
offered  in  the  curriculum,  and  showing  corresponding  irregularity 
in  the  development  of  the  mathematical  courses.  An  exact  resume 
of  the  changes  is,  of  course,  impossible.  The  following  sketch  is 
based  upon  the  changes  in  twenty  of  the  leading  institutions  and  is 
quite  characteristic  of  the  general  development. 

i)  CHANGES  IN  ENTRANCE  REQUIREMENTS. 

About  1850  some  algebra  began  to  be  required  for  entrance,  by 
1870  several  of  the  leading  institutions  required  one  year  and  by 
1908  &$%  have  made  a  requirement  of  one  and  one-half  years  of 
algebra.13  As  early  as  1855  a  very  few  institutions  required  plane 
geometry  for  entrance,  and  by  1875  there  were  still  very  few  requir- 
ing it,  while  it  was  quite  generally  required  in  1880.  The  University 
of  Michigan  and  Stevens  Institute  were  the  first  to  require  trigo- 
nometry, which  they  did  in  1890.  Since  then  more  and  more  have 
made  it  a  required  subject  until  now  iS%  do  so.13  A  few  institutions 

13  Table  of  entrance  requirements  Chap.  II,  p.  43. 


HISTORICAL  SUMMARY.  I5 

required  solid  geometry  in  the  early  eighties  and  between  1885  and 
1900  a  majority  of  the  institutions  made  this  a  requirement,  since 
which  time  there  has  been  little  change  up  to  the  present.13 

2)  COLLEGIATE  CURRICULUM. 

As  more  work  was  required  for  entrance  a  proportionate  amount 
of  a  more  advanced  nature  was  added  to  the  curriculum.  The  total 
time  given  to  mathematics  was  at  first  about  one  year  but  has  been 
increased  by  easy  stages  until  now  about  two  full  years  are  given  to 
it.  Trigonometry  was  either  required  for  entrance  or  formed  a  part 
of  the  college  work  in  every  course  organized.  Analytic  geometry 
has,  to  an  extent,  also  been  a  part  of  nearly  every  course.  The  time 
devoted  to  it  has  been  more  than  doubled  and  its  position  in  the  cur- 
riculum changed  from  the  last  part  of  the  second  year  or  first  part 
of  the  third  year  to  the  first  year.14  The  calculus  was  made  a  part 
of  some  engineering  schedules  as  early  as  1855  but  it  came  late  in  the 
course,  was  taken  only  for  one-third  or  one-half  of  a  year  and  did 
not  enter  into  the  work  of  the  course  to  any  extent.  The  increase 
in  the  attention  devoted  to  the  calculus  has  been  more  gradual  than  in 
any  other  subject,  save  that  of  algebra.  The  time  has  been  greatly 
increased,  a  whole  year  now  being  given  to  it  by  most  institutions, 
and  it  has  been  placed  earlier  in  the  curriculum ;  namely,  in  the  sec- 
ond year. 

3)  GENERAL  TREND. 

The  general  trend  of  late  years  has  been  to  require  one  and  one- 
half  years  of  algebra,  plane  and  solid  geometry  for  entrance  :15  and 
to  complete  algebra,  trigonometry  and  analytic  geometry  in  the  first 
year,  with  the  calculus  the  second  year.16  The  ratio  of  the  time  given 
to  each  subject  \vill  be  found  in  Chapter  II. 

u  Table  of  entrance  requirements.     Chap.  II,  p.  43. 
14  Table  of  collegiate  subjects.     Chap.  II,  p.  44. 
"  Table  of  entrance  requirements.     Chap.   II,  p.  43. 
M  Table  of  collegiate  mathematics.     Chap.  II,  p.  44. 


!6  MATHEMATICS  FOR  ENGINEERS. 

VIII.    PROGRESS  OF  INSTRUCTION  IN  MATHEMATICS. 

It  is  not  our  purpose  here  to  review  the  progress  of  mathemat- 
ical instruction  in  the  United  States."  During  the  early  years  of  the 
engineering  colleges  there  was  a  great  deal  of  poor  teaching,  owing 
to  the  fact  that  instructors  were  not  especially  prepared  in  mathe- 
matics. The  spirit  of  scientific  instruction,  the  learning  to  do  by 
doing,  which  has  been  adopted  extensively  by  the  professional  de- 
partments has  not  received  nearly  so  much  recognition  by  the  instruc- 
tors of  mathematics.  But  little  has  been  done  along  the  line  of 
formulating  and  presenting  the  work  in  courses  especially  adapted 
and  correlated  to  the  professional  subjects,  and  that  little  only  recent- 
ly.18 So  far  slight  attention  has  been  paid  to  the  training  of  instruct- 
ors as  teachers. 

IX.     SUMMARY. 

The  apprenticeship  system  and  the  Military  Academy  furnished 
all  of  the  engineering  education  until  about  1835.  With  the  found- 
ing of  the  Rensselaer  School  in  1824  began  the  formative  period  of 
the  engineering  colleges  which  lasted  until  about  the  time  of  the 
Civil  War.  The  courses  were  not  highly  specialized  and  were  most- 
ly variations  for  the  last  two  years  of  the  existing  courses  in  the 
literary  institutions.  There  was  no  stability  in  the  work  in  mathe- 
matics, either  as  to  time  allotted,  courses  offered  or  their  position  in 
the  schedule.  Although  the  National  Government  had  quite  early 
made  provision  for  military  and  naval  instruction  it  was  not  until 
1862  that  it  recognized  the  need  of  technical  instruction  along  civil 
lines,  by  the  passage  of  the  Land  Grant  Act.  Since  that  time  insti- 
tutions of  various  grades  have  been  established.  Contrasted  with  the 
changes  of  the  formative  period  those  of  the  present  have  been 
continuously  progressive ;  increasing  the  quantity  as  well  as  the  qual- 
ity of  the  work  in  mathematics,  placing  the  mathematical  courses  ear- 
lier in  the  curriculum  so  as  to  make  them  precede  most  of  the  techni- 
cal work,  with  the  gradual  evolution  of  the  present  schedule,  while 
close  correlation  of  the  work  in  mathematics  with  that  in  the  techni- 
cal subjects  has  but  lately  been  taken  up. 

1T  For  a  comprehensire  discussion  of  this  phase  see  Florian  Cajori,  The  Teaching  and 
History  of  Mathematics  in  the  United  States,  Government  Printing  Office,  1890,  pp.  98- 
CO& 

"Chap.  III. 


CHAPTER  II. 

ENGINEERING  COLLEGES  IN  THE  UNITED  STATES 

IN  1902. 

I.    CONTENTS. 

The  present  chapter  presents  in  condensed  form  data  bearing 
on  the  mathematical  instruction  of  freshmen  engineers  in  the  various 
institutions  in  the  United  States  offering  one  or  more  courses  in  engi- 
neering. It  includes  all  institutions  granting  a  degree  for  work  in 
engineering  and  graduating  a  class  of  at  least  six  from  such  courses, 
as  compiled  from  the  list  contained  in  the  Report  of  the  Commission- 
er of  Education  for  1908.  The  names  of  the  institutions  are  arrang- 
ed by  states  in  alphabetical  order. 

II.    SOURCES  OF  DATA. 

The  information  for  the  following  table  has  been  obtained 
largely  from  the  current  catalogs  of  the  various  institutions.  When 
this  information  was  lacking  or  given  in  such  a  manner  as  to  require 
considerable  interpretation,  a  letter  was  sent  to  the  institution  in 
question,  asking  for  the  missing  data  or  for  corrections  of  the  inter- 
pretations made.  If  no  corrections  were  received  the  conclusions 
were  assumed  to  be  correct  and  recorded  as  such  in  the  table.  In  a 
few  instances  it  has  been  impossible  to  obtain  the  necessary  data 
thru  either  catalogs  or  letters,  and  hence  parts  of  the  table  have 
necessarily  been  left  blank.  While  every  effort  has  been  made  to 
guard  against  errors  in  making  this  table,  it  is  too  much  to  expect 
that  some  have  not  crept  in.  It  is  only  hoped  that  the  errors  are 
not  grave  ones  and  that  no  institution  has  been  seriously  misrep- 
resented. 


1 8  MATHEMATICS  FOR  ENGINEERS. 

III.    EXPLANATIONS  OF  TABLES. 

The  following  explanations  are  necessary  to  an  understanding 
of  the  tables  on  pages  21  to  41. 

a)  Column  i  is  self  explanatory. 

b)  Column  2  gives  the  date  at  which  the  first  engineering  courses 
of  the  various  institutions  were  established. 

c)  Column  3  gives  a  list  of  the  courses,  as,  Civil  Engineering, 
Electrical  Engineering,  etc.,  offered  by  these  institutions  at  the  pres- 
ent time.    The  following  abbreviations  are  used : 

C.  E.,  Civil  Engineering. 
Mi.,  Mining  Engineering. 
Ch.,  Chemical  Engineering. 
Sani.,  Sanitary  Engineering. 
A.  E.,  Architectural  Engineering. 
A.,  Architecture. 
E.  E.,  Electrical  Engineering. 
M.  E.,  Mechanical  Engineering. 
Irrig.,  Irrigation  Engineering. 
Mun.,  Municipal  Engineering. 
N.  A.,  Naval  Architecture. 
Met.,  Metallurgy. 

d)  Column  4  contains  the  entrance  requirements  in  mathematics. 
The  statements  found  in  the  various  catalogs  have  occasionally  had 
to  be  modified  before  being  recorded  here,  since  in  some  instances 
neither  the  time  devoted  to  the  subject  nor  the  ground  covered  was 
definitely  stated  and  because  some  fixed  system  of  units  is  essential 
for  comparison.  The  usual  high  school  unit,  which  requires  at  least 
four  recitations  per  week  of  forty-five  minutes  each  for  thirty-six 
weeks,  was  the  one  selected  with  the  following  time  limit :  algebra  to 
quadratics,  one  year;  elementary  algebra  completed,  one  and  one- 
half  year;  plane  geometry,  one  year;  solid  geometry,  one  half  year; 
and  trigonometry,  one-half  year.  This  system  of  counting  conforms 
quite  closely  to  the  recommendation  of  the  Committee  on  Entrance 
Requirements  given  before  the  Society  for  the  Promotion  of  Engi- 


DATA  ON  ENGINEERING  COLLEGES.  ig 

neering  Education  in  lo/n,1  and  1902.*  If  no  definite  entrance  re- 
quirements were  stated  either  as  to  time  or  as  to  work,  deductions 
had  to  be  drawn  from  the  statement  of  the  courses  offered  in  mathe- 
matics for  the  freshman  year.  Wherever  elective  courses  in  entrance 
mathematics  are  found  the  minimum  requirement  is  given  and  the 
elective  courses  placed  in  the  freshman  year  of  the  college  work. 

e)  Column  5  states  for  which  course  (Civil  Engineering,  Me- 
chanical Engineering,  or  the  like)  the  data  of  the  succeeding  seven 
columns  are  tabulated.    This  was  found  necessary  as  no  one  course 
was  given  by  all  of  the  institutions.    To  preserve  as  much  uniformity 
as  possible,  the  Mechanical  Engineering  course,  whenever  given,  has 
been  scheduled. 

f)  Column  6  contains,  in  condensed  form,  the  mathematical 
work  of  the  curriculum.    This  is  all  required  except  that  (as  men- 
tioned above)   a  few  of  the  institutions  will  accept  for  entrance 
some  of  the  courses  scheduled  under  the  freshman  year.  For  the 
sake  of  uniformity  the  time  devoted  to  each  subject  has  been  reduced 
from  term,  semester  or  year  hours  to  that  of  the  actual  number  of 
recitation  hours ;  again  for  the  sake  of  uniformity  and  also  to  sim- 
plify the  work  very  considerably,  thirty-six  weeks  have  been  selected 
as  the  length  of  the  school  year. 

g)  Columns  7  to  12  inclusive  contain  the  number  of  hours  given 

1  "a"  Elementary  Algebra. 

1)  To   quadratics — The   four   fundamental   operations   for   rational   algebraic    expres- 
sions,   factoring,    highest    common    factor,    lowest    common    multiple,    complex    fractions, 
equations  of  the  first  degree  of  one  or  more  unknowns,  radicals,  fractional  exponents. 

2)  Quadratics  and  beyond — Quadratics   in   one   or   more   unknowns,   ratio   and  pro- 
portion,   progressions,    elements    of   permutations    and    combinations,    Binomial    Theorem 
with  integral  exponents  and  the  use  of  logarithms. 

"b"     Advanced  Algebra. 

To  include  the  elementary  treatment  of  infinite  series,  undetermined  coefficients,  Bi- 
nomial Theorem  for  fractional  exponents,  theory  of  logarithms,  determinants  and  the 
elements  of  the  theory  of  equations. 

Plane  Geometry. 

To  include  original  exercises  and  numerical  problems. 

Solid  Geometry. 

To  include  properties  of  straight  lines,  planes,  dyhedral  and  polyhedral  angles,  poly- 
hedrons, inclined  prisms,  pyramids,  regular  solids,  cylinders,  cones  and  spheres,  spherical 
triangles  and  measurement  of  surfaces  and  solids. 

Comittee  on  Entrance  Requirements,  Proceedings  of  the  Society  for  the  Promotion 
of  Engineering  Education,  vol.  IX,  p.  267. 

J  The  next  year  it  reported  the  following  change  desirable :  that  permutations  and 
logarithms  be  omitted ;  that  imaginaries  should  be  emphasized.  Ibid.,  vol.  X,  p.  200. 

The  part  marked  "Elementary"  under  algebra  is  that  for  the  one  and  one-half 
years'  high  school  work,  with  the  exception  of  the  use  of  logarithms  which  are  usually 
taught  with  trigonometry.  The  "Advanced"  is  the  usual  first  year's  college  work. 


20  MATHEMATICS  FOR  ENGINEERS. 

to  mathematics  (M),  professional  subjects  (P),  semi-professional 
subjects  (Sp),  language  (L),  miscellaneous  subjects  (Mi),  and  the 
total  (T).  The  hours  are  computed  as  for  column  six.  The  number 
in  column  seven  is  merely  the  sum'  of  the  number  of  hours  record- 
ed for  the  various  subjects  in  the  one  immediatelv  preceding  it. 
In  some  catalogs  surveying  and  mechanics  are  placed  under  the  De- 
partment of  Mathematics.  Whenever  it  has  been  possible  to  differ- 
eniate  these  from  the  purely  mathematical  subjects  they  are  counted 
as  professional.  This  has  been  done  to  preserve  uniformity,  as  in 
most  of  the  institutions  these  subjects  are  taught  by  the  professional 
departments.  Semi-professional  subjects  include  physics,  geology, 
chemistry,  astronomy  and  mineralogy.  Some  of  these  become  pro- 
fessional work  when  a  course  other  than  Mechanical  Engineering  has 
been  scheduled.  The  hours  given  under  language  include  English. 

The  general  rule  is  to  give  few  electives  except  where  the  work 
in  engineering  is  a  modification  of  a  literary  course.  Such  electives 
are  scheduled  under  miscellaneous  subjects.  Unless  otherwise  men- 
tioned the  time  required  to  complete  these  courses  is  four  academic 
years. 

/i)  Column  13  gives  the  percentages  of  the  total  time  which 
each  institution  devotes  to  mathematics. 


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a  « 

42  MATHEMATICS  FOR  ENGINEERS. 

V.    COLLECTED  RESULTS  OF  THE  TABLES. 

The  facts  given  in  detail  in  the  tables  are  collected  below  in 
summary  form  for  certain  important  phases  of  the  work. 

i)  GROWTH  OF  ENGINEERING  ACTIVITY. 

The  increase  in  the  number  of  institutions  offering  courses  in 
engineering  from  the  founding  of  the  United  States  Military  Acad- 


emy  in  1802  until  1908  is  shown  in  the  accompanying  graph.  Dates 
are  plotted  as  abscissae  and  the  total  number  of  institutions  giving 
courses  in  engineering  as  ordinates.  The  population  curve  for  the 
United  States  covering  this  period  is  also  plotted  to  the  same  axes. 


DATA  ON  ENGINEERING  COLLEGES. 


43 


2)  ENTRANCE  REQUIREMENTS.       (»')  Time  for  Various  Subjects. 

The  following-  table  gives  the  numerical  data  of  the  entrance 
requirements  of  the  one  hundred  and  thirty  institutions  recorded  on 
pages  21  to  41  as  presented  in  collected  form: 

No.  of  Institutions. 

Arithmetic    1 1 

Algebra. 

None I 

One-half  year   i 

One  year   14 

One  and  one-half  years 95 

Two  years 14 

Three  years  I 

Plane  Geometry. 

None 9 

One-half  year   3 

One  year 114 

Solid  Geometry. 

None 39 

One-half  year  87 

Trigonometry. 

One-half  year   24 

All  work  in  mathematics  (thru  the  calculus)  i 

Whether  plane  trigonometry  is  required  for  admission  or  accept- 
ed as  one  of  the  entrance  electives,  the  time  asked  for  it  is  in  all 
cases  one-half  year.  Nine  of  the  institutions  requiring  trigonometry 
for  entrance  give  a  review  of  it  during  the  college  course.  The  time 
required  for  the  completion  of  plane  geometry  is  uniformly  one  year 
and  for  the  completion  of  solid  geometry  one-half  year.  The  re- 
quirements in  algebra  vary  greatly  both  as  to  time  and  as  to  work, 
a  fact  which  is  greatly  to  be  deplored  as  it  is  the  lack  of  preparation 
in  algebra  that  gives  the  student  of  engineering  his  greatest  difficul- 
ties, as  will  be  fully  discussed  a  little  later.  It  will  be  seen  that  the 
time  requirements  vary  from  nothing  at  all  to  three  full  years ;  the 
great  majority  or  73%,  however,  require  one  and  one-half  years. 
Of  the  fourteen  which  require  two  years  seven  include  college  alge- 
bra and  the  remaining  seven  a  half  year's  review  during  the  last 
year  of  the  preparatory  course. 


44 


MATHEMATICS  FOR  ENGINEERS. 


(»')  Requirements  of  a  Majority  of  the  Institutions. 

The  preceding  table  shows  that  the  great  majority  of  the  insti- 
tutions require  for  entrance  one  and  one-half  years  of  algebra,  one 
year  of  plane  geometry  and  one-half  year  of  solid  geometry. 

% 
3)     CURRICULUM — (i)  Position  of  Subject. 

The  following  table  shows  the  number  of  institutions  in  which 
the  various  subjects  are  given  at  the  stage  in  the  curriculum  that 
is  named : 

No.  of  Institutions. 

Trigonometry. 

For  entrance   13 

Completed  first  year   94 

Completed  second  year  1 1 

Completed  third  year I 

Algebra. 

All  for  entrance 13 

Completed  first  year   95 

Completed  second  year  12 

Completed  third  year 2 

Analytic  Geometry. 

All  for  entrance  I 

Completed  first  year   59 

Completed  second  year  59 

Completed  after  second  year 4 

Calculus. 

All  for  entrance  I 

Completed  first  year   2 

Begun  first  year,  completed  second  year 14 

All  in  second  year 75 

Completed  after  second  year 31 

Differential  equations  are  mentioned  as  separate  courses  by  20 
of  the  institutions,  and  theory  of  equations  by  7.  As  a  rule  the 
latter  is  included  in  the  work  scheduled  under  algebra  and  does  not 
receive  special  mention.  The  theory  of  least  squares  is  included  by 
4.  Only  39  mention  offering  spherical  trigonometry,  the  others 
either  specify  plane  trigonometry  only  or  give  no  hint  as  to  the 


DATA  ON  ENGINEERING  COLLEGES. 


45 


contents  of  the  course.    In  the  same  way  only  40  specify  giving  solid 
analytic  geometry. 

The  discrepancies  in  the  above  table  are  due  to  two  things: 
first,  that  no  data  could  be  obtained  from  7  of  the  institutions,  and 
secondly,  that  n  of  the  institutions  give  special  courses3  in  mathe- 
matics from  which  the  subjects  mentioned  could  not  very  readily  be 
differentiated. 

(M)  General  Trend  of  Position  of  Subjects  in  Curriculum. 

From  the  above  table  and  that  regarding  entrance  requirements 
on  page  43,  we  see  that  there  is  a  general  trend  towards  giving  trig- 
onometry, algebra  and  analytic  geometry  in  the  first  year  of  the  col- 
lege course,  and  all  of  the  calculus  in  the  second  year. 


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3  Chap.  III. 


46  MATHEMATICS  FOR  ENGINEERS. 

(t/i)  Comparison  of  Time  for  Various  Subjects. 

The  accompanying  cuts,  which  are  self  explanatory,  give  a  com- 
parison of  the  time  devoted  to  the  various  subjects,  trigonometry, 
algebra,  analytic  geometry  and  the  calculus  by  the  various  institu- 
tions recorded  on  pages  21  to  41. 

4)  PERCENTAGE  OF  TIME  GIVEN  TO  MATHEMATICS. 

The  percentages  of  the  total  time  of  instruction  in  the  four  col- 
legiate years  given  to  mathematics  are  exhibited  in  the  graph  on 
page  45- 

5)  DISCREPANCIES  AND  THEIR  CAUSES. 

The  great  variety  in  the  number  of  hours  recorded  in  the  col- 
umns shows  the  lack  of  uniformity  among  the  engineering  colleges 
of  the  United  States.  Some  of  the  more  noticeable  ones  will  here 
be  briefly  considered.  Where  entrance  requirements  in  mathematics 
are  low  the  time  devoted  to  the  collegiate  work  is  correspondingly 
large.  The  wide  range  in  the  amount  of  time  given  miscellaneous 
subjects  is  due  to  the  fact  that  a  varying  degree  of  importance  is 
attached  to  culture  as  compared  with  professional  subjects,  or  more 
often  that  the  engineering  courses  are  in  varying  degrees  modifica- 
tions of  the  general  science  courses.  The  great  variation  in  the  num- 
ber of  hours  recorded  under  "P,"  "Sp"  and  especially  under  "T,"  is 
accounted  for  in  two  ways.  The  first  and  real  factor  is,  that  different 
relative  values  are  given  to  recitations,  lectures,  laboratory  and 
industrial  periods.  By  "industrial  periods"  is  meant  such  work  as 
drawing,  shop-work,  etc.,  which  requires  no  outside  preparation. 
The  following  four  ratios  of  equivalence  of  the  different  kinds  of 
work  are  found  in  use: 


No.  of  hours  in 
i  

Mathematics 
i  

Laboratory 

2       

Industrial 
.  .  "? 

ii   

i  

2       

.2V2 

iii  

i  

..2V2.. 

..2V2 

iv  . 

.  i.  . 

-7 

.  .2 

In  the  majority  of  cases  the  institution  which  used  the  first 
ratio  recorded  the  lowest  number  of  hours  under  "professional" 


DATA  ON  ENGINEERING  COLLEGES. 


47 


Percentage  of  Total  Ma.£-/ie/n<zr<ca.t 
Instruction  given  tro  Trigonometry. 


x& 


i 


fr 


Percentage  ofTotral 
Instruction,  given  to  College  fit $e bra. 


48 


MATHEMATICS  FOR  ENGINEERS. 


3t 


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12 


Cv? 


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of  Totnl  Mat/t em at 
Instruction  $<v£H  fro  flnafyt-t' 


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of  Total 
Instruction  give*  to  e/te   Co.tculti$. 


DATA  ON  ENGINEERING  COLLEGES. 


49 


subjects,  which  increased  for  the  second,  third  and  fourth  ratios  in 
that  order  of  sequence.  The  second  cause  for  variation  is  merely 
a  clerical  one,  arising  from  insufficient  data  in  the  various  catalogs. 


VI.    PERCENTAGE  OF  TIME  GIVEN  VARIOUS  GROUPS 

OF  WORK. 

One  further  computation  was  made:  the  percentages  were 
computed  for  each  of  the  divisions  "mathematics,"  "professional," 
"semi-professional,"  "language"  and  "miscellaneous"  by  adding  the 
total  time  recorded  for  each.  The  results  were  as  follows : 

Professional 52.3 %  Semi-professional    15.0% 

Mathematics , .  13.7%  Language    11.0% 

Miscellaneous 8.0% 


VIL     SUMMARY. 

There  are  one  hundred  and  thirty  instituions  giving  degrees  for 
work  in  engineering,  one  hundred  and  twenty-one  of  which  have 
been  organized  since  1860;  the  general  entrance  requirements  are 
one  and  one-half  years  algebra,  one  year  plane  geometry  and  one- 
half  year  of  solid  geometry  with  trigonometry  required  by  iS%  of 
the  institutions;  the  general  trend  of  collegiate  mathematics  is  to 
complete  trigonometry,  algebra  and  analytic  geometry  the  first  year 
and  to  devote  the  second  year  to  calculus,  except  that  a  few  institu- 
tions are  beginning  to  introduce  the  calculus  into  the  first  year; 
there  is  great  variation  in  the  allotment  of  time  to  the  various  de- 
partments and  especially  to  the  department  of  mathematics  where  the 
allotment  varies  from  $%  to  23%  of  the  total  time;  mathematics 
receives  only  one-fourth  as  much  time  as  the  professional  studies 
and  only  slightly  more  than  those  in  language. 


CHAPTER  III. 


RECENT  MODIFICATIONS  IN  THE  WORK  IN, MATH- 
EMATICS FOR  STUDENTS  OF 
ENGINEERING. 

I.  CONTENTS  OF  THE  CHAPTER. 

As  already  noted1  some  of  the  engineering  colleges  of  the  United 
States  have  discarded  the  formal  division  of  the  work  in  mathematics 
into  subjects  separated  by  sharply  defined  lines,  and  some  institu- 
tions have  added  new  features,  such  as  laboratory  work  in  curve 
tracing,  special  courses  in  computation,  etc.  A  treatment  of  this 
modified  work  will  be  the  object  of  the  present  chapter. 

II.    GENERAL  NATURE  OF  THE  MODIFIED  WORK  IN 
MATHEMATICS. 

This  departure  from  the  old  lines  aims  to  arrange  the  work 
in  mathematics  for  the  student  of  engineering  to  fit  his  special  needs, 
and  is,  as  will  be  seen,  in  full  conformity  with  the  so-called  "Perry 
Movement."2  In  1902  Prof.  Moore  advocated  an  immediate  reform 
in  the  teaching  of  elementary  collegiate  mathematics  so  as  to  bring  it 

1  Chap.  II,  pp.  21  to  41. 

*  It  may  be  best  to  give  Prof.  Perry's  idea  of  the  necessary  reform  in  the  teaching 
of  mathematics  in  his  own  words :  "Newton  employed  geometrical  conies  in  his  astron- 
omical studies  and  so  mechanics  was  developed,  and  therefore  it  is  that  every  young  en- 
gineer must  study  mechanics  thru  astronomy,  and  he  dares  not  think  of  differential 
calculus  till  he  has  finished  geometrical  conies.  The  young  applier  of  physics,  the 
engineer,  needs  a  teaching  of  mathematics  which  will  make  his  mathematical  knowl- 
edge part  of  his  mental  machinery,  which  he  will  use  as  readily  and  as  certainly  as  a 
bird  uses  its  wings — ."  John  Perry,  On  the  Teaching  of  Mathematics,  Nature,  vol.  26.  p, 
319- 


RECENT  MODI  PICA  T1ONS  IN  MA  THEM  A  TICS.  5 1 

within  the  realm  of  reality.3  Such  modifications  have  been  worked 
out  in  a  number  of  engineering  institutions  where  the  object  has 
been  to  build  up  a  logical  course  in  mathematics  based  upon  the 
future  needs  of  the  student.4  These  modifications  are  as  yet  in  the 
evolutionary  stage  but  have  as  their  object  the  treatment  of  mathe- 
ematics  as  a  scientific  "weapon."5 


III.     SOURCES  OF  DATA. 

Twenty-two  of  the  institutions  considered  in  Chapter  II  are 
found  to  have  established,  to  a  greater  or  less  extent,  such  modified 
work  in  mathematics.  In  order  to  secure  more  complete  informa- 
tion regarding  this  work  than  is  found  in  the  catalogs  a  questionaire 
letter  was  sent  to  each  institution,  uniformly  containing  the  follow- 
ing questions  together  with  such  others  as  were  suggested  by  the 
separate  catalogs. 

8  "Just  as  the  secondary  schools  should  begin  to  reform  without  waiting  for  the 
improvement  of  the  primary  schools,  so  the  elementary  collegiate  courses  should  be  mod- 
ified at  once  without  waiting  for  the  reform  of  the  secondary  schools."  " —  a  feeling 
that  mathematics  is  indeed  itself  a  fundamental  reality  of  the  domain  of  thought  and 
not  merely  a  matter  of  symbols  and  arbitrary  rules  and  conventions."  Presidential 
Address  at  the  Annual  Meeting  of  the  American  Mathematical  Society.  Bull.  Am. 
Math.  Soc.,  1903,  pp.  1-24. 

*  "Some    technical    schools    have    made    radical    changes    in    the    mathematical    work 
of  the  college  course.  Those  which  have  worked  out  the  matter  independently  have  arrived 
at  very  similar  results.     The  changes  consist  for  the  most  part  of  modifications  that  nat- 
urally  follow   from   the   movement   in    the   secondary    schools.      The   tendency   is   to   blot 
out  entirely  the  present  lines  of  distinction  between  algebra,   trigonometry   and   analytic 
geometry.     These  changes  have  involved,  not  only  the  early  and  constant  use  of  graph- 
ical methods  and  abundant  use  of  numerical  data,  but  have  also  included  the  early  con- 
sideration and  application  of  vector  analysis.     Our  most  treasonable  act  is  the  deposing 
of   the  conic    sections,    which   have   been   the   reigning   family    for   so    many   years,    from 
their   exalted   place   in   our   course   of   study.      This    gives   an   opportunity    for   the   more 
abundant   enrichment   of  the  course   and   a   better  comprehension   of   those   things   which 
the  student  of  applied  science  needs.     Such  a  course  not  only  conforms  more  nearly  to  the 
actual   needs  of  the  students,   but  it  also  has   the  advantage   of  being  more   logical   and 
scientific   than   our   old   course.      As   a   matter   of   fact    all   that   the   engineering   student 
learns  in  his  usual  course  in  mathematics  is  a  simple  comprehension  of  the  properties  of 
the   algebraic    functions   of   a   real   variable.      Indeed   we   might   define    the   work   of   the 
engineering  student,  during  his  first  two  years  of  mathematics,  as  the  study  of  the  expo- 
nential function  of  the  real  variable."    Chas  S.  Slichter,  The  Improvement  of  the  Fresh- 
man Year  of  Mathematical  Instruction  in  Technical  Schools,   Proceedings  of  the  Society 
for  the  Promotion  of  Engineering  Education,  vol.   14,  p.   146. 

*  " — a  most  powerful  weapon  with  which  to  unlock  the  mysteries  of  Nature."  John 
Perry,  Address  before  the  Section  of  Educational   Science  of  the  British  Association,  at 
Glasgow,  1902. 


52  MATHEMATICS  FOR  ENGINEERS. 

i )  QUESTIONAIRE  LETTER. 

(i)  In  what  way  does  the  work  differ  from  that  usually  given 
in  a  general  science  or  arts  course  ? 

(11)  What  is  the  material  used  and  from  what  source  is  it 
obtained  ? 

(in)     What  is  the  particular  object  sought? 

(iv)  When  was  this  instruction  with  special  reference  to  engi- 
neering begun,  and  what  suggested  the  same? 

While  some  replies  were  quite  brief,  others  were  very  compre- 
hensive and  described  fully  the  work  given.  No  replies  were  receiv- 
ed from  seven  of  the  institutions  written  to,  in  which  cases  reference 
will  be  made  to  the  statements  found  in  the  catalogs  wherever  usable. 

IV.     SYNOPSIS   OF   PRINCIPAL   FEATURES   OF 
MODIFIED  WORK. 

The  following  synopsis  shows  for  twenty  of  the  above  institu- 
tions separately  the  principal  features  of  the  instruction  ascertain- 
ed as  just  specified.  The  statements  are  arranged  according  to  sim- 
ilarity of  their  principal  features,  and  are  followed  by  a  short  sum- 
mary. 

INSTITUTION  No.  i.    COHERENT  COURSE  OF  ESSENTIAL  ELEMENTS. 

The  reply  to  the  letter  of  inquiry  suggested  the  consultation  of 
"A  Course  in  Mathematics,"  by  Woods  and  Bailey,  the  first  volume 
of  which  is  used  as  a  text  during  the  freshman  year. 

It  may  be  well  to  quote  from  the  preface  in  regard  to  the  aims 
and  objects  of  the  authors :  "This  book  is  the  first  volume  of  a  course 
in  mathematics  designed  to  present  in  a  consecutive  manner  an 
amount  of  material  generally  given  in  distinct  courses  under  the  var- 
ious names  algebra,  analytical  geometry,  differential  and  integral  cal- 
culus, and  differential  equations In  arranging  the  material, 

however,  the  traditional  division  of  mathematics  into  distinct  subjects 
is  disregarded,  and  the  principles  of  each  subject  are  introduced  as 
needed  and  the  subjects  developed  together.  The  objects  are  to  give 
the  student  a  better  grasp  of  mathematics  as  a  whole,  and  of  inter- 
dependence of  its  various  parts,  and  to  accustom  him  to  use,  in 


RECENT  MODIFICATIONS  IN  MATHEMATICS. 


53 


later  applications,  the  methods  best  adapted  to  the  problem  in  hand. 
At  the  same  time  a  decided  advantage  is  gained  in  the  introduction  of 
the  principles  of  analytic  geometry  and  the  calculus  earlier  than  is 
usual.  In  this  way  these  subjects  are  studied  longer  than  is  other- 
wise possible,  thus  leading  to  greater  familiarity  with  their  methods 
and  greater  freedom  and  skill  in  their  application." 

(*)  Study  and  Use  of  Curves. 

Accordingly  the  work  in  mathematics  in  this  institution  appears 
to  be  as  follows :  A  few  preliminary  lessons  are  devoted  to  determ- 
inants, considering  only  evaluation  of  determinants,  their  use  in 
the  solution  of  linear  simultaneous  equations,  eliminants  and  the 
testing  of  equations  for  common  roots. 

Some  work  on  the  different  kinds  of  numbers  and  functions, 
introduces  a  graphic  study  of  polynomials.  After  a  short  considera- 
tion of  derivatives  of  polynomials  there  follows  a  discussion  of 
curves,  considerable  time  being  devoted  to  conies  but  only  as  a  species 
of  curves  in  general,  where  such  questions  as  tangents,  normals, 
diameters,  cusps,  asymptotes,  branches,  intersections  of  curves  and 
systems  of  curves  are  taken  up.  The  general  equation  of  the  second 
degree  is  not  taken  up  at  all.  Polar  and  parametric  representation  of 
curves  are  taken  up  as  separate  topics  and  applied  to  problems  for 
which  they  are  most  servicable.  Solid  analytic  geometry  is  deferred 
until  the  second  year. 

(«)  Early  Introduction  of  the  Calculus. 

Derivatives  of  polynomials  are  introduced  quite  early  in  the 
course  thru  the  idea  of  the  slope,  and  are  immediately  applied  to 
problems  on  tangents,  and  maxima  and  minima.  A  more  complete 
treatment  follows  the  study  of  curves  as  mentioned  above,  where 
simple  problems  of  physics  are  also  considered.  Quite  a  number  of 
transcendental  functions  are  studied  towards  the  close  of  the  course, 
which  ends  with  a  brief  treatment  of  curvature,  evolutes  and  invo- 
lutes of  curves. 

INSTITUTION  No.  2.     MAXIMUM  USABLE  MATHEMATICS  IN  FIRST 
YEAR. 

In  this  institution  a  course  called  "An  Introduction  to  Modern 
Mathematics"  is  given  the  first  year  the  aim  of  which  is  a  "desire 
to  start  the  student  with  as  much  mathematics  as  possible  in  the 


54 


MATHEMATICS  FOR  ENGINEERS. 


freshman  year  so  that  he  can  grasp  work  required  in  the  profession- 
al schools  as  early  as  possible."  It  includes  "much  that  was  former- 
ly taught  under  the  head  of  Theory  of  Equations,  combined  with  an 
early  introduction  to  the  elements  of  analytic  geometry  and  calculus, 
— while  not  anticipating  the  calculus  to  any  extent."  "A  Course  in 
Mathematics"  is  also  used  as  a  text  here  and  supplemented  by  Went- 
worth's  Analytic  Geometry",  especially  for  problems  in  solid  analytic 
geometry. 

(*')   Origin  of  Course. 

This  course  was  introduced  in  1907  as  many  of  the  students 
came  well  prepared  in  trigonometry  and  higher  algebra  so  that  such 
a  course  could  be  taken  advantageously,  both  as  to  time  and  prepa- 
ration. A  rigorous  examination  in  trigonometry  must  be  passed  by 
all  students  admitted  to  it.  The  class  is  divided  into  sections  based 
upon  the  ability  of  the  students. 

INSTITUTION    No.    3.      MATHEMATICAL    SCHEDULE   A    COHERENT 
WHOLE. 

The  catalog  states,  "The  aim  of  the  instruction  in  mathematics 
is  to  present  the  subject  so  that  the  student  may  obtain  a  thoro 
working  knowledge  of  these  principles  which  he  needs  to  know 
when  he  becomes  an  engineer.  It  is  recognized  that  such  knowledge 
can  best  be  obtained  by  an  exercise  of  the  observational  faculties  of 
the  student,  by  treating  the  subject  as  one  coherent  whole  rather  than 
a  series  of  more  or  less  disconnected  subjects,  and  by  frequent  appli- 
cation of  the  principles  taught  to  problems  in  engineering.  For 
this  reason  laboratory  methods  are  sometimes  employed — and  the 
subjects  taught  are  so  arranged  that  each  is  a  help  in  the  development 
of  the  others.  We  give  the  calculus  in  the  freshman  year  in  order 
that  the  men  may  early  become  acquainted  with  the  calculus  method 
so  as  to  use  the  subject  more  intelligently  in  the  many  cases  which 
arise  during  the  sophomore  year,  in  physics,  kinematics,  etc." 

(*')  College  Algebra. 

The  freshmen  take  five  hours  of  algebra  during  the  first  half 
of  the  year,  comprising  "a  review  of  equations;  plotting  of  curves 
from  equations ;  a  comprehensive  treatment  of  surds,  imaginaries,. 
ratio,  proportion,  variation;  the  progressions;  permutations;  com- 


RECENT  MODIFICATIONS  IN  MATHEMATICS.  55 

binations ;  determinants ;  binomial  theorem  for  positive  integral 
powers  of  the  binomial ;  logarithms ;  partial  fractions ;  methods  of 
approximations  to  the  roots  of  equations ;  building  up  of  equations 
from  the  properties  of  the  curves.  A  special  feature  of  the  course  is 
the  introduction  of  a  large  number  of  problems  and  curves  similar 
to  those  met  by  the  engineers  in  actual  practice  in  order  to  drill  the 
student  on  many  algebraic  operations  peculiar  to  engineering,  espe- 
cially those  for  logarithmic  computations  which  are  not  found  in  the 
ordinary  algebra.  By  stating  the  actual  engineering  problem,  with 
the  physical  law,  we  try  and  do,  to  a  great  extent,  arouse  the  student's 
interest  in  the  subject  of  college  algebra,  a  study  that  is  too  often 
found  dry  and  uninteresting." 

(«)  Analytic  Geometry  and  Calculus. 

Five  hours  per  week  are  given  during  the  second  semester  to 
analytic  geometry  and  the  elements  of  the  calculus,  when  are  consid- 
ered: "Transformation  of  coordinates;  a  systematic  treatment  of 
the  circle,  parabola,  ellipse  and  hyperbola ;  limits ;  the  ordinary  rules 
for  differentiation  with  application  to  curve  plotting,  rates,  maxima 
and  minima;  the  fundamental  forms  of  integration  with  easy  appli- 
cations to  problems  in  plane  areas;  those  subjects  in  college  algebra 
and  analytical  geometry  which  lend  themselves  best  to  the  calculus 
treatment."  In  analytical  geometry  the  straight  line  and  loci  prob- 
lems receive  the  principal  consideration;  for  conies  only  questions 
of  vertices,  foci  and  directrices  are  taken  up.  Problems  on  tangents 
are  treated  by  the  calculus  and  are  not  restricted  to  conies.  The 
various  forms  of  differentiation  are  taken  up  together  with  their 
applications. 

(in)  Material  Used. 

The  texts  on  algebra  and  trigonometry  are  supplemented  by 
numerous  mimeographed  problems  as  mentioned  above,  of  which  the 
following  are  types : 

\    r. 

«)  Snout* 


-  2}  EVALUATE  (3.i6)'142  and  'f*  x3-*5  X  .3* 

A/.004  X  .00032 

3)  SOLVE  THE  EQUATION  Js*~+ 


MATHEMATICS  FOR  ENGINEERS. 


Given  that  (an  expression  in  X)  /*  means  the  value  of  the  ex- 
pression when  X  =  a  minus  its  value  when  X  =  b,  and  given  e  = 
2.71828,  evaluate 

4)  LOG  (sec  X  -f  tan  X)l  ~f 

fo 

5)  SOLVE  FOR  X  AND  Y  J43y~*3 

/3.4y+x=  16* 


6)  FIND  LOG 


V, 
i8l/2 

7)  EXPAND  THE  DETERMINANT 


iii 

357 
984 


8)  SOLVE    ^l_^4  =  x^_^Z 

X-io      X-6       X-7      X-9 

Use  logarithms  in  the  following  problems  involving  computa- 
tions. 

9)  Within  the  elastic  limit,  the  expression  of  a  spiral  due  to  a 
weight  suspended  to  one  end  is  proportional  to  the  weight.    A  spring 
originally  18.23"  l°ng  measures  20.35"  when  a  weight  of  3.2  oz.  is 
attached ;  what  will  be  its  length  when  the  weight  is  4.5  oz.  ? 

10)  Within  the  elastic  limit,  the  extension  of  a  rod  under  tension 
varies  directly  as  the  product  of  the  length  and  the  tension,  and  in- 
versely as  the  area  of  the  cross-section.     If  a  round  wrought  iron 
rod,  3'  long,  .625"  in  diameter,  is  stretched  .0057"  by  a  tension  of 
6000  Ibs.,  how  much  will  a  bar  of  wrought  iron  io/  long  and  of  rec- 
tangular cross-section  .25"  by  .375"  be  stretched  by  a  tension  of 
560  Ibs.? 

Several  problems  are  given  under  each  of  the  conditions  as 
stated  above. 

INSTITUTION  No.  4.    A  MODIFIED  COHERENT  SCHEDULE  OF  SUB- 
JECTS. 

Beginning  with  the  work  in  algebra  and  continuing  thru  the 
calculus  the  work  of  this  institution  is  arranged  as  a  systematic 
whole,  while  not  wholly  disregarding  the  division  into  the  subjects 
algebra,  analytic  geometry,  differential  and  integral  calculus. 


RECENT  MODIFICATIONS  IN  MATHEMATICS. 


57 


INSTITUTION  No.  5.    ARRANGEMENT  OF  MATHEMATICS  COURSES  FOR 
FUTURE  WORK. 

The  freshman  work  begins  with  a  review  of  surds,  imaginaries 
and  quadratic  equations,  followed  by  those  principles  needed  in 
analytic  geometry.  The  whole  mathematical  schedule  is  arranged 
with  view  to  the  work  taken  by  the  students  in  their  third  and  fourth 
years  covering  such  subjects  as  mechanics,  hydraulics,  machine  de- 
sign, bridges,  etc.  This  arrangement  was  reported  as  meeting  with 
the  approval  of  the  professional  departments  as  well  as  with  that 
of  the  instructors  of  mathematics,  the  only  difficulty  being  that  of 
finding  suitable  text-books.  Trigonometry  is  given  as  a  formal 
course  in  the  first  year  and  all  that  is  given  of  the  calculus  constitutes 
the  second  year's  work. 

INSTITUTION  No.  6.    COMBINED  COURSE  IN  ALGEBRA  AND  TRIGONO- 
METRY. 

A  course  in  algebra  and  trigonometry  is  given  the  first  half  of 
the  freshman  year  in  which,  "the  work  in  algebra  deals  with  topics 
supplementing  the  work  in  trigonometry ;"  later  in  the  course  trigo- 
nometry in  turn  is  used  in  the  solution  of  equations. 

(»')  Mathematics  as  a  Tool. 

As  to  the  objects  and  the  handling  of  the  course  in  mathematics 
an  instructor  reports : — "We  lay  stress  upon  the  theory  but  insist 
upon  familiarity  with  processes  and  manipulations  of  results.  For 
instance,  in  the  study  of  Taylor's  Theorem,  we  do  not  go  into  the 
remainder  form  of  the  theorem,  but  lay  stress  upon  the  actual  use  of 
the  theorem  in  the  expansion  of  functions.  In  the  integral  calculus 
we  lay  less  stress  upon  the  derivation  of  the  reduction  formulas  than 
upon  the  use  of  the  tables  of  integration.  We  develop  the  calculus 
as  a  tool  rather  than  as  a  science  for  its  own  sake." 

INSTITUTION  No.  7.    ALGEBRA  THRUOUT  THE  COURSE  IN  MATHE- 
MATICS. 

No  formal  course  is  given  in  algebra  but  the  subject  is  distrib- 
uted thruout  the  whole  course  in  mathematics.  When  a  topic  is 
reached  that  calls  for  any  principles  in  algebra  which  are  new  to  the 
students  or  which  present  difficulties,  such  principles  are  taken  up 
and  elucidated. 


58  MATHEMATICS  FOR  ENGINEERS. 

INSTITUTION  No.  8.    EARLY  CALCULUS. 

A  course  in  analysis  is  given  the  second  half  of  the  first  year 
in  which  are  treated  some  of  the  advanced  topics  of  algebra  and  an 
introduction  to  the  calculus.  This  is  followed  by  a  formal  course 
in  analytic  geometry,  where  the  principles  of  the  former  courses 
are  applied. 

INSTITUTION  No.  9.    MATHEMATICS  AS  BASIS  OF  TECHNICAL  TRAIN- 
ING. 

The  mathematical  courses  are  conducted  by  means  of  well 
chosen  text-books,  very  little  supplementary  material  being  used. 
Special  attention  is  given  to  practical  problems  which  are  interesting 
to  the  students.  "The  ability  to  understand  and  apply  mathematical 
processes  readily  is  the  aim,  and  to  this  end  special  emphasis  is  laid 
upon  two  things:  elucidation  of  the  principles  and  drill  upon  their 
application,  as  furnishing  the  only  sure  basis  for  a  thorough  technical 
and  professional  training." 

INSTITUTION  No.  10.    APPLIED  CONTRASTED  WITH  PURE  MATHE- 
MATICS. 

The  reply  from  this  institution  was  especially  clear  in  stating 
the  object  of  their  special  work  in  mathematics  and  its  variation 
from  that  held  in  view  for  the  art  students.  To  quote:  "The  object 
we  have  in  view  is  to  give  the  students  facility  in  using  mathematics 
intelligently  in  solving  engineering  problems.  In  the  A.B.  course, 
on  the  other  hand,  we  aim  to  give  the  student  a  broad  view  of  the 
field  of  mathematics  as  a  branch  of  human  knowledge  and  some  of  its 
big  results.  In  brief,  these  aims  animate  our  methods." 

For  example  the  engineer  "has  to  study  the  elements  of  accu- 
racy in  testing ;  this  demands  partial  derivatives,  etc."  For  the  A.B. 
student,  "Partial  derivatives  come  under  the  general  head  of  space 
differentiation  and  involve  a  grip  of  the  real  meaning  of  calculus  in 
its  widest  sense." 

"I  know  that  there  are  some  that  contend  that  the  engineer 
should  have  this  broad  insight  into  mathematical  ideas,  but  the 
thing  is  not  feasible.  He  is  primarily  interested  in  material  construc- 
tion, and  he  cannot  focus  his  attention  on  two  things.  He  is,  and 
ought  to  be,  concerned  with  effective  construction.  In  handling  this, 


RECENT  MODIFICATIONS  IN  MATHEMATICS.  59 

mathematics  is  to  him  a  different  thing  from  what  it  is  to  one  who 
is  interested  in  the  harmonies  of  numbers  and  space.  I  do  not  believe 
there  is  an  engineering  mathematics,  but  I  do  believe  there  is  an  engi- 
neer's use  of  mathematics. 

"Therefore  our  object  primarily  is  to  teach  the  engineer  to  use 
mathematics.  To  do  this  he  must  recognize  the  type  of  his  problem, 
must  understand  the  method  of  solving  that  type,  and  must  under- 
stand the  construction  of  the  method ;  just  as  he  must  know  in  any 
case  what  tool  to  use,  how  to  use  it,  how  to  modify  it,  to  adapt  it  or 
even  to  make  it."  The  results  are  reported  as  "on  the  whole  very 
good." 

(i)  Problems  from  "Real  Material." 

Whenever  possible  real  material  is  used  for  problems  which  is 
obtained  from  "laboratories,  kinematics,  graphic  statics,  etc. — we 
aim  to  supplement  rather  than  duplicate  the  problems  in  these  cours- 
es." The  following  are  two  illustrations  of  the  problems  used : 

"An  elliptical  ceiling  in  a  church  has  a  20  foot  span  and  a 
6  foot  rise  at  the  center.  Rafters  run  tangent  to  it  at  angles 
of  30°  and  60°  to  the  horizontal,  with  a  horizontal  on  top. 
Find  where  their  intersections  are,  their  lengths,  and  where  the 
two  at  60°  strike  the  wall.  Draw  same  to  scale." 

For  algebra:  "The  diameter  of  a  solid  wooden  column  is 
given  by  the  formula  d2— isd  (F/P— i)— 700  (F/P — i) 
=  o;  P  -=  ultimate  strength  (lb./in.2),  F  =  crushing  strength 
(lb./in.2) .  For  white  oak  F  =  5000,  P  =  4586.  Find  d." 

(«')  Laboratory  Work. 

The  students  make  models  of  the  solids  mostly  used  and  do  a 
great  deal  of  drawing  to  scale  with  pencil,  using  "cross-section  paper, 
polar  coordinate  paper  ruled  both  for  degrees  and  radians,  loga- 
rithmic, sine  ruled,  etc."  They  complete  about  one  hundred  illustra- 
tions of  various  curves  and  graphic  solutions  in  this  way. 

INSTITUTION  No.  n.    NATURE  OF  APPLIED  MATHEMATICS. 

"The  study  of  Mathematics,  as  pursued  at  this  school,  is  chiefly 
for  the  purpose  of  acquiring  a  working  knowledge  of  its  use  in  the 
subsequent  studies  of  engineering,  physics,  and  chemistry,  and  not 
merely  as  a  component  part  of  a  general  education."  The  chief  dif- 


60  MATHEMATICS  FOR  ENGINEERS. 

ference  from  the  usual  work  given  the  arts  students  lies,  says  one 
instructor,  in  "the  omission  of  topics  which,  we  have  found,  do  not 
make  any  impression  on  the  student's  mind,  or  add  to  his  skill  in 
analysis  and  in  manipulation,  and  are  of  no  particular  use  in  his  sub- 
sequent study.  I  find  it  useless  to  teach  such  things  as  undetermined 
coefficients,  partial  fractions,  permutations  and  combinations  in  the 
Freshman  year.  I  give  plenty  of  time  to  the  calculus  and  teach  the 
things  when  they  are  needed,  so  that  the  student  may  see  what  they 
are  good  for,  and  have  some  chance  of  retaining  them." 

(i)  Graphic  Algebra  and  Trigonometry. 

The  freshmen  take  "Graphic  Algebra,  Curve  Tracing  and  Alge- 
braic Analysis"  for  the  first  three  months.  Besides  the  work  on 
graphics  the  most  striking  features  are  "convergency  of  infinite 
series;  use  of  infinite  series  in  approximation  calculations;  errors 
of  observations ;  methods  of  least  squares."  The  next  three  months 
are  devoted  to  acquiring  "a  knowledge  of  the  trigonometric  functions 
in  analysis  and  in  shortening  computations."  The  remaining  three 
months  are  devoted  to  the  solution  of  the  oblique  triangle,  "limits, 
expansion  of  functions  in  series,  application  of  De  Moivre's  Theo- 
rem," and  spherical  triangles. 

(«)  Analytic  Geometry  and  Calculus. 

Analytic  Geometry  is  taken  the  first  three  months  of  the  second 
year.  "The  matter  and  methods  are  intended  to  aid  the  student  in 
his  subsequent  reading  of  technical  literature,  and  in  solving  the 
problems  which  arise  in  his  work  in  Mechanics  and  Physics." — The 
work  in  the  calculus  considered  from  the  engineer's  standpoint  occu- 
pies the  remainder  of  the  year  and  considers  differentiation,  proper- 
ties of  tangents  and  normals  and  problems  in  maxima  and  minima, 

(Hi)  Problems  from  Engineering. 

"No  special  reference  is  made  to  engineering  problems  in  the 
first  year,  and  it  has  been  considered  unprofitable  to  use  material 
from  engineering  practice  or  engineering  literature  without  first 
carefully  preparing  it  for  our  students.  It  is  necessary  to  avoid  the 
appearance  of  teaching  engineering  or  physics  but  to  make  it  clear  to 
the  student  that  we  are  merely  teaching  the  kind  of  mathematics 
useful  in  engineering. — I  never  got  satisfactory  results  until  I  cut 
loose  from  all  text-books,  that  is,  class-room  use  of  text-books." 


RECENT  MODIFICATIONS  IN  MATHEMATICS.  61 

INSTITUTION  No,  12.    MATHEMATICAL  POWER  AND  FORMAL  COURSE. 

The  aims  and  results  of  this  institution  are  very  well  described 
by  an  instructor:  "The  objects  I  have  in  view  with  students  are  two- 
fold ;  first,  to  develop  in  them  mathematical  power,  including  insight 
and  initiative,  variety  and  strength  of  attack,  all  dependent  of  course 
not  upon  guess  work  but  upon  clear,  solid  analysis;  second,  to  give 
them  control  over  the  formal  part  of  mathematics  as  a  tool  and  its 
methods.  The  results  have  been  the  deepest  possible  interest  in  the 
subject  by  multitudes  of  students  and  a  disposition  to  place  mathe- 
matics study  foremost  in  the  favor  and  devotion  of  students." 

(i)  Problems  from  Engineering. 

In  analytic  geometry  and  the  calculus  special  emphasis  is  given 
problems  in  engineering  in  order  to  bring  before  the  student  an 
early  realization  of  the  great  importance  mathematics  plays  in  engi- 
neering practice.  Cross-section  paper  is  widely  used  in  the  fresh- 
man work  and  the  students  taught  to  deduce  the  results  directly  from 
the  graphs.  Texts  are  made  the  basis  of  the  work  but  problems  are 
drawn  from  a  variety  of  sources. 

INSTITUTIONS  Nos.  13  AND  14.    ENGINEERING  PROBLEMS. 

In  two  of  the  institutions  the  mathematical  studies  are  taken  up 
with  special  reference  to  their  connection  with  mechanics.  Problems 
are  selected  from  engineering  data,  the  students  being  taught  how  to 
obtain  and  make  use  of  the  same  successfully  together  with  the  use 
of  tables  and  computing  devices. 

INSTITUTION  No.  15.    EXTRA  PROBLEMS  FOR  ENGINEERS. 

This  institution  adds  a  sufficient  number  of  problems  to  the 
course  given  to  the  students  in  engineering  to  double  the  time  for 
the  same  subject  given  to  the  arts  students.  The  bulk  of  these  prac- 
tical problems  relate  to  surveying  while  considerable  attention  is 
given  to  the  selection  of  trigonometric  equations  as  a  grounding  for 
future  analytic  work. 

INSTITUTION  No.  16.    DIVISION  INTO  Two  CLASSES:  ANALYSIS  AND 
COMPUTATION. 

Two  courses  are  given  simultaneously  the  first  half  of  the 
freshman  year.  The  analytic  side  of  algebra,  trigonometry  and  ana- 


62  MATHEMATICS  FOR  ENGINEERS. 

lytic  geometry  is  considered  thoroly  in  what  is  designated  as  "A 
Course  in  Analysis."  The  other  is  "A  Course  in  Computations;" 
the  various  computations  arising  in  connection  with  work  not  involv- 
ing the  calculus  are  taken  up  and  computating  instruments  such  as 
planimeters,  slide  rules,  etc.,  are  studied  and  used. 

(«)  Contents  of  Course  in  Computation. 

This  work  is  given  in  two-hour  periods;  the  first  hour  being 
used  by  the  instructor  in  a  lecture,  after  which  the  students  work 
at  computations  under  the  instructor's  direction.  The  reason  for 
this  second  course  is  that  it  is  believed  " — that  in  the  elementary 
branches  of  the  subject  the  student  should  be  taught  'systematic  com- 
putation,' which  can  best  be  done  under  personal  supervision  of  the 
instructor  and  his  assistants." 

(n)  Freshman  Calculus. 

The  second  half  of  the  year  is  devoted  to  an  elementary  treatise 
on  differentiation  and  integration,  Ransom's  "Freshman  Calculus" 
being  used.  The  purpose  of  the  course  is  " — to  provide  the  student 
of  science  or  engineering  very  early  in  his  course  with  a  familiarity 
with  the  fundamental  conceptions  and  methods  of  the  calculus  in  as 
far  as  they  are  of  use  in  the  elementary  study  of  the  physical 
sciences." 

INSTITUTION  No.  17.    PROBLEMS  AND  COMPUTATION  DEVICES. 

The  problems  thruout  the  mathematical  courses  are  selected 
from  actual  questions  arising  in  connection  with  "mines,  stamp 
mills,  power  plants,  etc.,"  in  the  vicinity  of  the  institution.  Permu- 
tations and  combinations  or  other  subjects  of  only  a  disciplinary 
value  are  excluded.  Tables,  slide  rules,  planimeters,  etc.,  are  used 
in  computations. 

INSTITUTION  No.  18.     COMPUTATIONS. 

A  short  course  for  attaining  accuracy  is  given  in  the  freshman 
year.  Degrees  of  accuracy,  short  methods,  use  of  tables  and  calcu- 
lating instruments  are  the  chief  features. 


RECENT  MODIFICATIONS  IN  MATHEMATICS.  63 

INSTITUTION  No.  19.    COMPUTATION — PURPOSE. 

The  general  idea  permeating  the  work  is  to  insure  a  working 
knowledge  of  mathematics  to  the  students.  The  particular  feature 
is  a  course  of  two  hours  for  three  months  of  the  freshman  year  in 
mensuration  and  logarithms,  which  is  a  computation  course  in  prob- 
lems of  physics,  mechanics  and  engineering. 

INSTITUTION  No.  20.    LABORATORY  WORK. 

The  special  course  in  mathematics  is  confined  to  one  two  hour 
laboratory  period  per  week  taken  in  connection  with  one-half  year 
each  of  algebra  and  analytical  geometry  during  the  freshman  year. 
A  laboratory  period  is  also  given  to  juniors  taking  analytical  mechan- 
ics. It  has  been  necessary  to  discontinue  laboratory  work  in  cal- 
culus on  account  of  a  lack  of  time. 

(i)  Instruments  and  Material  Used. 

All  work  is  done  in  pencil  upon  "ten  by  ten  to  the  inch  cross- 
section  paper"  which  is  cut  to  "six  and  three-fourths  by  ten,  and  ten 
by  thirteen  and  one-half  inches."  Drawing  tables  accommodating 
four  students  are  used. 

(«)  Problems  Considered. 

"The  problems  cover  the  plotting  of 
Y  =  sinX,    Y  =  tanX,    Y  =  secX, 

or  the  co-functions;  which,  together  with  plotting  in  polar  coordi- 
nates, gives  some  review  and  exercise  in  trigonometry.  In  connec- 
tion with  algebra,  problems  in  graphical  solution  of  single  equations 
and  pairs  of  simultaneous  equations  are  solved.  Along  with  ana- 
lytic geometry  are  given  further  exercises  in  curve  plotting,  by 
points,  illustrating  among  other  things,  inflections  (not  the  actual 
location),  asymptotes,  nodes,  and  cusps,  and  to  some  extent  the 
relation  between  form  of  equation  and  form  of  curve  (this  being  a 
difficult  matter  with  the  student).  We  also  plot  certain  systems  of 
conies.  During  the  differential  calculus  I  have  had  series  plotted,  as  r 

Y  =  X,    Y  =  X— *L,   Y  =  X—  *.+  £.,   Y  =  sinX, 
|_3_  |_3_      [5_ 

illustrating  convergency.  Another  style  of  problem  has  to  do  with 
curvature ;  as  to  draw  a  parabola  and  the  circle  of  curvature  for  the 
vertex,  and  then  draw  the  circle  of  curvature  for  some  other  point. 


64  MATHEMATICS  FOR  ENGINEERS. 

Examination  of  such  drawings,  carefully  made  by  the  student,  is 
quite  instructive.  Another  exercise  is  to  plot  Y  =  f  (X)  and  sev- 
eral of  its  derivatives." 

(HI)   Workings  of  the  Course. 

The  instructor  in  charge  of  each  laboratory  section  gives  it  his 
personal  attention.  He  also  " — makes  up  the  problems,  all  members 
of  the  class  working  on  the  same  problem  at  about  the  same  time. 
This  of  course  results  in  some  little  copying,  but  on  the  other  hand, 
the  advantages  of  comparison,  one  with  the  other  and  the  suggestions 
one  student  gets  from  another  outweigh  that."  For  calculus  they  use 
the  "laboratory  period  as  a  practice  period  in  the  integral  calculus, 
beginning  the  subject  before  finishing  the  differential,  the  students 
working  at  the  board,  with  the  instructor  passing  around  answering 
questions,  offering  suggestions  and  correcting  the  work.  In  the  lab- 
oratory period  the  fundamental  formulae  were  gotten  thru  with  by 
the  time  of  finishing  differential  calculus,  but  the  laboratory  period 
was  retained  thru  the  integral  calculus." 

(iv)  Object  of  Course. 

The  object  is  " — to  supplement  and  broaden  the  analytical  part 
of  trigonometry,  advanced  algebra  and  analytics.  It  has  been  my  ex- 
perience in  assigning  curves  to  be  plotted  at  home  to  have  all  degrees 
of  accuracy  and  slovenly  work  presented  and  very  little  good  work. 
The  student  also  appears  to  regard  the  graph  as  an  unnecessary  pic- 
ture or  carricature  of  the  equation,  a  thing  to  be  drawn  in  tree 
hand  by  an  artist  of  the  impressionist  school."  The  poor  preparation 
of  the  student  is  assigned  as  another  reason  for  giving  this  work, 
because  the  personal  supervision  of  the  instructor  for  a  long  period 
is  thought  to  be  beneficial. 

(v)  Results. 

As  to  results  and  working  of  the  course  we  read :  "Our  classes 
are  small,  about  twenty  for  freshmen  and  ten  for  sophomore  mathe- 
matics. This  makes  it  possible  to  provide  tables  and  blackboard 
room  for  working  a  whole  class  at  once.  Of  course  the  laboratory 
work  takes  more  time  of  the  instructor  and  we  have  also  found 
another  difficulty ;  that  of  finding  sufficient  time  during  the  day  for 
students  to  take  all  their  necessary  laboratory  work." 

The  only  objections  raised  were  those  of  extra  demands  upon 


RECENT  MODIFICATIONS  IN  MATHEMATICS.  65 

the  teaching  force  and  some  difficulty  in  arranging  the  schedule  of 
the  students.  After  a  four  years'  trial  the  plan  was  reported  a  suc- 
cess and  well  liked  by  the  students.  It  was  further  said  to  aid  " — in 
giving  not  a  little  practice  in  numerical  calculations  and  for  this 
we  encourage  the  use  of  the  tables,  as  of  squares  and  cubes.  This  is 
made  the  occasion  of  mentioning  the  slide  rule." 

V.     SUMMARY. 

I ) Very  few  institutions  mentioned  the  date  at  which  they  had  first 
taken  up  this  modified  work  in  mathematics.  No  date  earlier 
than  1900  was  given. 

2)  Number  of  institutions  that  give  trigonometry 16 

3)  Number  of  institutions  that  give  algebra all 

4)  Number  of  institutions  that  give  plane  analytical  geometry  .  .  .all 

5)  Number  of  institutions  that  give  solid  analytic  geometry  .... 

first  year    7 

second  year   10 

6)  Number  of  institutions  that  give  the  calculus  first  year 7 

second  year  15 

7)  Number  of  institutions  that  give  the  calculus  as  a  separate 

course 14 

8)  Number  of  institutions  that  give  the  calculus  in  connection 

with  other  work   8 

9)  Number  of  institutions  that  give  courses  more  or  less  of  a 

laboratory  nature   9 

10)  Material  Used: 

i)  General  text-books  used  only 4 

ii)  Text-books  arranged  for  engineering  students 3 

iii)   Specially  prepared  pamphlets,  with  or  without  texts. .  2 
iv)  Text-books  supplemented  by  problems  and  exercises 
taken  from  laboratories  and  professional  depart- 
ments   4 

n)  Main  Features. 

Problems  given  much  more  prominence  than  in  the  mathemati- 
cal courses  for  arts  students. 

Omission  of  useless  and  uninteresting  topics  in  algebra — the  use 
of -especially  prepared  lists  of  problems  in  algebra. 
Computation  courses  under  supervision. 
Laboratory  periods  for  work  in  graphs. 


66  MATHEMATICS  FOR  ENGINEERS. 

Topics  arranged  so  as  to  make  one  coherent  course  without 
division  into  subjects. 

Aims  to  give  an  early  working  knowledge  of  mathematics,  espe- 
cially of  the  calculus. 

Selection  of  topics  which  will  form  a  dieect  part  of  the  future 
work  in  engineering  subjects. 

12)  Causes  for  the  Establishment  of  These  Modified  Courses. 
Good  preparation  of  entering  students, — a  cause   for  giving 

more  advanced  work. 

Poor  preparation  of  entering  students, — a  cause  for  giving  lab- 
oratory work. 

Agitation  in  the  journals  and  at  the  meetings  of  the  various  soci- 
eties. 

To  attain  the  requirements  of  the  professional  departments. 

13)  Objects  of  the  Course. 

To  create  interest  in  algebra. 

To  ground  students  in  the  solution  of  problems. 

To  teach  the  value  of  the  graph  and  its  use  in  the  interpretation 
of  results. 

To  introduce  the  calculus  early  so  as  to  equip  the  students  as 
soon  as  possible  with  this  weapon,  and  at  the  same  time  giving  them 
more  practice  in  its  use  and  for  a  longer  time. 

To  give  a  working  knowledge  of  mathematics  and  ability  to  use 
it. 

To  familiarize  the  students  with  the  mathematics  they  will  need 
for  their  future  technical  work. 

14)  Results. 

Only  a  few  of  the  letters  received  gave  any  expression  as  to  the 
results  attained  in  these  modified  courses.  All  that  did  so,  however, 
were  highly  in  favor  of  the  new  mode  of  procedure.  Only  one  in- 
stance was  found  of  a  return  to  the  more  formal  treatment.  That 
was  the  omission  of  laboratory  work  in  the  calculus,  necessitated  by 
a  lack  of  time. 


CHAPTER  IV. 

CURRENT  THOUGHTS  ON  VITAL  QUESTIONS. 

I.    MODE  OF  OBTAINING  THE  DATA. 

To  obtain  a  survey  of  current  thought  on  certain  important 
questions  651  copies  of  the  following  questionaire  letter  were  sent 
out. 

For  convenience  the  questions  were,  as  far  as  possible,  made 
answerable  by  yes  or  no,  or  by  numerical  annotation,  even  at  the 
cost  of  making  them  more  narrow  and  formal  than  would  other- 
wise have  been  desirable.  In  order  not  to  make  (2),  (4)  and  (7) 
complete  tables  of  contents  for  these  various  subjects  and  thus  ren- 
der them  too  large  to  be  readily  answered,  topics  a  under  each  had  to 
include  a  larger  number  of  operations  than  would  otherwise  have 
been  desired. 

i)  QUESTIONAIRE  LETTER. 

UNLESS  OTHERWISE  STATED  THE  FOLLOWING  WILL 

RELATE  TO  THE  COURSES  IN  MATHEMATICS 

FOR  FRESHMEN  STUDENTS  IN 

ENGINEERING. 

1.  Is  it  advisable  to  require  entrance  examinations  in  mathe- 
matics of  all  students ? 

2.  Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
topics  which  are  fundamental  to  a  course  in  trigonometry: 

a.  Derivation  and  manipulation  of  formulae. 

b.  Solution  of  triangles  by  use  of  natural  functions. 

c.  Use  of  logarithms  in  solution  of  triangles  and  other 
computations. 

d.  Problems  applying  the  solution  of  triangles. 

e.  Higher  trigonometry — hyperbolic  functions,   DeMoiv- 
re's  Theorem,  etc. 


68  MATHEMATICS  FOR  ENGINEERS. 

j.  In  what  particular  topics  of  trigonometry  do  students  show 
themselves  most  deficient  after  the  freshman  year? 

4.  Number  i,  2,  3,  etc,,  in  order  of  importance  the  following 
topics  which  are  fundamental  to  a  course  in  college  algebra: 

a.  Review  of  surds,  exponents,  quadratics,  etc. 

b.  Series. 

c.  Binomial  Theorem. 

d.  Permutations  and  combinations. 

e.  Graphs. 

f.  Determinants. 

g.  Theory  of  Equations. 

5.  In  what  particular  topics  of  algebra  do  students  show  them- 
selves most  deficient  after  the  freshman  year? 

6.  Should  the  object  of  analytic  geometry  be  (a)  to  learn  the 
characteristics  and  theory  of  some  class  of  curves,  as  conies,  or  (b) 
to  learn  a  new  mathematical  language,  using  any  suitable  problems 
in  curves  or  mechanics  as  the  medium  for  the  study  of  this  language  f 

7.  Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
topics  which  are  fundamental  to  a  course  in  analytic  geometry: 

a.  Algebraic  relations  for  intersection  of  loci,  tangency, 
etc. 

b.  Loci  problems. 

c.  Right  line  and  circle. 

d.  Conies. 

e.  Higher  curves,  as  cycloids,  etc. 

f.  Coordinates  of  points  in  space  with  application  to  sim- 
ple loci. 

8.  In  what  particular  topic  of  analytic  geometry  do  stdents 
ivho  have  completed  a  course  in  that  subject  show  themselves  most 
deficient  f 

p.  Would  it  be  advisable  to  introduce  a  working  knowledge  of 
elementary  differentiation  and  integration  preceding  or  simultane- 
ously with  the  course  in  analytic  geometry  f 

10.  What  is  the  remedy  for  (3),  (5),  and  (8)? 

11.  Is  it  advisable  to  treat  problems  met  with  in  actual  engi- 
neering work? 

12.  If  so,  to  what  extent  and  under  which  topics  of  (2),  (4), 
and  (7)? 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  69 

/j.     Is  a  short  course  in  computation  advisable? 

14.  Should  freshmen  engineers  be  taught  their  mathematics  in 
.separate  classes  or  together  with  those  taking  other  courses? 

15.  Why? 

16.  Are  special  texts  in  mathematics  for  engineering  students 
advisable? 

if.    How  often  should  written  quizzes  be  given? 

18.  Number  i,  2,  j,  etc.,  the  order  of  usefulness  of  the  follow- 
ing modes  of  conducting  classes  suggesting  any  further  modes  or 
combinations  of  these: 

a.  Assignment  of  work  from  text  without  any  previous 
explanation. 

b.  Assignment  of  work  from  text  with  previous  explana- 
tion. 

c.  New  work  explained,  the  student  taking  notes — with  or 
without  the  use  of  a  text.    Written  or  oral  quizzes  at 
regular  intervals. 

d.  Students  led  to  discover  new  principles  by  suggestions 
and  quizzes  on  the  old. 

e.  Each  student  working  independently,  going  as  fast  as 
he  is  able.    This  may  apply  to  the  entire  course,  to  each 
topic,  or  to  each  day's  work. 

f.  First  part  of  hoivr  devoted  to  the  class  as  a  whole;  sec- 
ond part  given  to  individual  work. 

g.  Subject  treated  from  the  laboratory  standpoint;  most 
of  the  work  being  done  in  the  class  room. 

/p.  Number  i,  2,  3,  etc.,  the  order  of  importance  of  the  fol- 
lowing pertaining  to  the  qualifications  of  the  engineering  student 
who  has  completed  the  courses  in  freshman  mathematics: 

a.  Skill  and  accuracy  in  computations. 

b.  Analysis  of  problems. 

c.  Interpretations  of  results  in  solution  of  problems. 

d.  Knowledge  and  use  of  equations. 

e.  Representation  of  physical  laws  by  means  of  graphs. 

.   20.    In  which  of  (19)  do  students  who  have  completed  their 
freshman  year  show  the  greatest  deficiency? 
21.     What  is  the  remedy? 


70  MATHEMATICS  FOR  ENGINEERS. 

22.  Designate  by  i,  2,  3,  etc.,  the  order  of  importance  for  any 
mathematical  topic  of: 

a.  Mastery  of  theoretical  matter  involved. 

b.  System  and  neatness. 

c.  Accuracy  in  computed  results. 

23.  Is  it  true,  as  has  been  said,  that  engineering  students  apply 
themselves  better  than  those  taking  a  literary  or  scientific  -course? 

2)  REPLIES. 

The  six  hundred  and  fifty-one  letters  sent  and  the  replies  receiv- 
ed are  classified  in  the  following  table : 

No.  No.  of         %  of 
Sent                Replies       Replies 
To  instructors  of  mathematics  and 
professional  courses,  represent- 
ing 92  institutions 544  204  37.5 

To  teachers  of  mathematics  especi- 
ally interested  in  the  question 20  13  65.0 

To  professional  engineers 87  20  23.0 

Total 651  237  36.4 

The  practical  side  is  represented  more  strongly  in  the  replies 
than  the  number  would  indicate,  since  several  of  the  instructors  of 
professional  subjects  are  also  practicing  engineers. 


II.    TABULATION  OF  DATA. 

A  summary  of  the  replies  to  these  questions  is  given  in  what 
follows.  The  questions  are  taken  up  in  order  and  separately.  Fol- 
lowing a  short  review  of  the  important  points  considered  in  the  ques- 
tion are  appended  some  of  the  most  suggestive  and  helpful  com- 
ments in  the  form  of  quotations.  The  discrepancy  which  will  often 
be  found  between  the  total  number  of  replies  received  and  that  re- 
corded for  any  particular  question  is  accounted  for  by  the  fact  that 
several  writers  omitted  replies  to  various  questions. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  71 

III.    ENTRANCE  EXAMINATIONS. 

i.  Is  it  advisable  to  require  entrance  examinations  in  mathe- 
matics of  all  students? 

Replies : 

Yes  130 

No  72 

Several  of  those  who  opposed  this  requirement  did  so  on  the 
ground  of  inexpediency  but  would  otherwise  have  favored  it,  while 
many  who  favored  it  voiced  the  same  objections;  namely,  (a)  that 
it  would  cause  serious  disturbance  in  the  accredited  secondary  schools 
and  (b)  that  if  the  rule  of  an  institution  is  to  receive  its  students 
from  the  secondary  schools  without  examination  it  would  not  be  fair 
to  the  other  departments  to  make  an  exception  in  favor  of  mathemat- 
ics. 

Few  complaints  were  found  of  poor  preparation  in  geometry 
but  a  very  large  number  in  algebra.  The  general  opinion  seemed  to 
be  that  something  stronger  than  a  mere  request  should  be  presented 
to  the  secondary  schools  for  better  preparation  in  algebra.  The  fol- 
lowing quotation  expresses  the  reasons  in  favor  of  entrance  exam- 
inations:—  "I  think  that  all  students  entering  engineering  courses 
should  be  required  to  take  entrance  examinations  for  the  following 
reasons:  ist,  to  find  their  deficiencies  in  order  that  these  may  be 
remedied  before  it  is  too  late;  2nd,  to  insure  that  the  high  school 
preparation  is  what  it  should  be ;  3rd,  to  insure  that  a  good  prepara- 
tion has  not  been  forgotten ;  4th,  to  discourage  men  deficient  in  math- 
ematical ability  from  undertaking  the  engineering  courses  without 
extra  preparation,  and  to  prevent  those  lacking  in  such  ability  from 
entering  such  a  course  at  all." 

i)  A  SUBSTITUTE  FOR  THE  ENTRANCE  EXAMINATION. 

One  method  which  is,  in  a  way,  a  compromise  between  these 
two  extremes  is  used  by  some  of  our  foremost  and  rising  institutions. 
This  plan,  as  outlined  in  a  letter  from  one  of  the  institutions  using 
it,  is  as  follows :  "Regarding  Entrance  Examinations  in  Mathematics, 


72  MATHEMATICS  FOR  ENGINEERS. 

I  would  say  that  it  seems  to  us  not  desirable  to  require  Entrance 
Examinations  in  the  case  of  a  State  University,  even  in  such  a  sub- 
ject as  Mathematics,  on  account  of  the  fact  that  the  University 
should  take  its  place  as  part  of  the  general  school  system  of  the  state. 
This  it  cannot  do  if  it  stands  aloof  from  these  schools,  and  accepts 
their  students  only  after  an  examination,  which  explicitly  refuses 
recognition  of  the  quality  of  the  work  done  in  the  secondary  schools. 

"We  recognize,  however,  that  the  students  entering  on  certifica- 
tion are  often  very  deficient  in  mathematical  knowledge.  In  order 
to  avoid  the  setting  of  examinations,  and  to  provide  some  basis  for 
weeding  out  the  poorest  of  these  students,  we  have  struck  upon  the 
following  scheme:  Some  two  weeks  or  more  after  the  opening  of 
school,  after  a  rapid  review  of  those  topics  in  elementary  algebra 
which  precede  quadratic  equations,  we  give  a  test  covering  this 
ground.  The  students  who  fail  in  this  test  are  not  allowed  to  con- 
tinue this  course.  We  do  not  remove  their  credits  for  entrance,  we 
do  not  even  require  them  to  take  up  the  study  of  elementary  algebra, 
but  we  make  it  rather  obvious  that  the  only  way  they  will  ever  suc- 
ceed in  this  course  is  by  a  thoro  review  of  that  work,  and  we  point 
out  to  them  that  an  opportunity  to  do  this  is  afforded  in  several  of 
the  Secondary  Schools  in  operation  in  this  city.  To  the  next  grade  of 
students,  those  who  do  not  utterly  fail,  but  are  weak  on  this  test,  we 
give  the  alternative  namely,  that  they  either  drop  the  course  at  once, 
or  that  they  continue  it,  together  with  a  review  of  elementary  alge- 
bra, in  the  training  school  of  the  Teachers'  College  of  his  University. 
We  believe  that  this  scheme  is  superior  to  the  scheme  of  entrance 
examinations,  and  highly  superior  to  the  scheme  of  direct  admission 
to  classes  on  certification  without  restriction." 

No  unfavorable  reply  was  received  from  any  one  who  had  ever 
tried  this  plan. 


IV.    SPECIFIC  NEEDS  AND  DEFICIENCIES. 

The  following  seven  questions  are  concerned  with  the  needs, 
deficiencies  and  remedies  pertaining  to  topics  in  the  subjects  general- 
ly taught  the  freshman  year;  logarithms,  trigonometry,  algebra,  de- 
terminants, etc. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  73 

i  TRIGONOMETRY — t)  Needs. 

2.     Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
topics  which  are  fundamental  to  a  course  in  trigonometry: 

a.  Derivation  and  manipulation  of  formulae. 

b.  Solution  of  triangles  by  use  of  natural  functions. 

c.  Use  of  logarithms  in  the  solution  of  triangles  and  in 
other  computations. 

d.  Problems  applying  the  solution  of  triangles. 

e.  Higher  Trigonometry — hyperbolic  functions,  DeMoiv- 

re's  Theorem,  etc. 


Replies  : 

I. 

II. 

III. 

IV. 

V. 

F. 

a 

147 

17 

21 

34 

i 

935 

b 

40 

63 

37 

63 

6 

695 

c 

33 

92 

54 

29 

i 

754 

d 

33 

40 

79 

55 

4 

676 

e  o  o  9  ii  158  207 

The  columns  of  the  above  table,  indicated  by  the  Roman  numer- 
als, refer  to  the  order  of  importance  assigned  to  the  various  topics 
mentioned  in  the  question.  The  number  in  any  column  opposite  any 
letter  tells  how  many  of  those  answering  the  question  gave  it  this 
particular  order  of  importance.  Thus  there  were  37  who  placed  b 
as  third  in  order,  11  who  placed  e  as  fourth,  etc.  The  numbers  in 
the  final  column,  marked  F,  are  obtained  by  adding  5  times  the  num- 
ber in  the  first  column,  4  times  that  in  the  second,  etc.,  as  for  a 
147x5  -f-  17x4  -f  21x3  -f-  34x2  +  ixi  =  935,  which  shows  the  rel- 
ative importance  assigned  to  a,  b,  c,  d,  e.  It  should  be  mentioned, 
however  that  a  large  number  thought  e  should  be  wholly  omitted. 

Comments: 

"In  trigonometry,  more  time  slhould  be  devoted  to  practical  problems  in 
plane  trigonometry;  much  less  to  spherical  trigonometry,  which  is  rarely 
used  even  by  civil  engineers.  It  is  singular  that  perhaps  more  students  fail 
in  trigonometry  than  in  other  branches;  as  it  is  comparatively  easy,  the 
system  of  instruction  must  be  at  fault." 

"Few  students  are  drilled  to  use  logarithms  intelligently,  rapidly  or  cor- 
rectly. Most  tables  of  logarithms  and  logarithmic  functions  in  American 
text-books  are  very  inconvenient  and  far  inferior  to  the  German  tables  in 


74  MATHEMATICS  FOR  ENGINEERS. 

arrangement,  and  instructors  seem  scarcely  to  consider  this  in  selecting  a 
text-book.  Hence  the  student  never  likes  to  use  logaritJims.  Then  he  is 
often  required  to  use  6  or  7  place  tables  for  computation  in  which  4  place 
tables  would  be  amply  sufficient." 

"Regarding  the  importance  of  the  topics  which  you  mentioned  in  trig- 
onometry I  would  say  that  the  order  appears  to  me  to  be  as  follows : 
b,  c,  d,  a.  I  have  omitted  e  entirely,  for  its  importance  is  highly  problemati- 
cal in  a  course  on  trigonometry  for  freshmen,  since  I  do  not  believe  that 
freshmen  have  any  use  for  this  matter,  and  I  do  not  believe  that,  they  can 
get  a  proper  grasp  of  it.  I  have  placed  b  first  simply  because  it  seems  to 
me  to  involve  the  true  spirit  of  trigonometry  as  a  whole.  If  the  student 
grasp  the  essential  notion  of  the  solution  of  right  triangles,  he  has  a  founda- 
tion for  the  whole  of  trigonometry.  I  place  c  next  since  the  question  of 
computation  by  logarithms  and  otherwise  is  the  main  use  to  which  the 
freshman  will  put  his  trigonometrical  knowledge.  I  place  d  next  since  the 
practical  problems  of  trigonometry  are  the  type  to  which  he  will  apply  this 
knowledge.  The  derivative  and  the  manipulation  of  formulae  are  important 
enough,  but  they  are  certainly  secondary  to  the  general  grasp  of  the  purpose 
of  elementary  trigonometry,  which  is  essentially  contained  in  the  first  three. 

"The  spirit  of  the  remarks  I  have  just  made  will  follow  through  all  I 
have  to  say  in  what  follows,  namely,  in  each  subject.  I  shall  regard  as  of 
first  importance  the  general  conception  of  that  subject,  next  the  particular 
means  of  putting  these  into  operation,  and  next  the  things  to  which  they  are 
applied." 

(»)  DEFICIENCIES. 

3.     In  what  topics  of  the  above  in  trigonometry  do  students 
show  themselves  most  deficient  after  the  freshman  year? 

Replies :  a  b  c  d  e 

126  22  33  34  12 

f~  Very  few  replies  reported  difficulties  under  more  than  one  topic 
given  in  question  (2).  It  is  interesting  to  note  that  55%  of  the  dif- 
ficulties reported  fall  under  a,  the  derivation  and  manipulation  of 
formulas.  A  few  of  the  letters  specified  particular  topics  under  a 
as  giving  the  greatest  amount  of  difficulty  as  follows : 

*)   50%  in  the  manipulation  of  formulae,  especially  in  identities, 
functions  of  2x  and  ^£x; 

2)  25%  in  inverse  functions; 

3)  I2l/i%  in  circular  functions; 

4)  i2l/2%  in  application  of  definitions. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  75 

Comments: 

"In  trigonometric  equations." 

"They  can  not  apply  definitions  to  rapid  solution  of  triangles." 

"The  greatest  weakness  I  find  in  the  student  having  finished  trigonom- 
etry is  an  inability  to  interpret  a  given  problem  in  the  trigonometrical  terms; 
that  is,  he  is  unable  to  see  the  triangles  which  must  be  solved." 

"Technical  engineering  courses  (undergraduate)  require,  as  a  rule,  little 
mathematics  beyond  the  elements.  Hence  in  these  courses  it  is  only  in  the 
elements  of  mathematics  that  students  show  either  weakness  or  strength. 
When  the  attempt  is  made  in  the  freshman  mathematics  courses  to  cover 
much  ground  including  'advanced'  work,  the  inevitable  result  with  a  large 
proportion  of  students  is  weakness  in  the  elements,  which  reveals  itself  in 
all  subsequent  work  involving  mathematics.  The  remedy,  as  far  as  a  rem- 
edy is  possible,  is  to  confine  the  work  in  the  freshman  and  sophomore  math- 
ematical subjects  mainly  to  the  elements  and  to  teach  these  as  efficiently  as 
possible." 

.2)  ALGEBRA     (i)  Fundamentals. 

4.  Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
•which  are  fundamental  to  a  course  in  college  algebra : 

a.  Review  of  surds,  exponents,  quadratics,  etc. 

b.  Series. 

c.  Binomial  Theorem. 

d.  Permutations  and  combinations. 

e.  Graphs. 

f.  Determinants. 

g.  Theory  of  Equations. 

Replies : 

a 

b 

c 

d 

e 

/ 

g  9        25        24        27        26        34        28        615 

The  key  for  (2)  also  applies  to  the  above  table.  A  few  replies 
gave  an  equal  rank  to  two  or  more  topics,  others  omitted  one  or  more 
as  non-essential  to  engineering  students.  Of  the  latter  /,  c,  and  d 
were  the  principal  ones  mentioned  and  in  that  order. 


I. 

II. 

III. 

IV. 

V. 

VI. 

VII 

F. 

177 

15 

ii 

3 

i 

o 

0 

1399 

2 

34 

42 

51 

26 

15 

10 

639 

19 

83 

56 

19 

15 

5 

o 

1032 

I 

4 

14 

34 

36 

36 

48 

465 

23 

52 

48 

33 

24 

8 

10 

895 

9 

4 

14 

14 

38 

60 

38 

426 

76  MATHEMATICS  FOR  ENGINEERS. 

Comments: 

"I  should  omit  the  subject  of  determinants  and  give  very  little  time  to 
permutations  and  combinations  on  a  course  in  engineering  algebra.  Graphic 
analysis  should  be  introduced  in  the  high  school  and  should  be  taken  up  in 
connection  with  all  of  the  topics  studied. 

"In  addition  to  the  subjects  mentioned  under  (4)  a  thorough  course  in 
Limits  and  Logarithms  is  essential  to  the  work  in  algebra.  I  believe  I  would 
class  these  two  subjects  as  of  greater  importance  than  the  Binomial  Theorem." 

"A  good  deal  of  the  work  which  we  often  attempt  to  do  in  advanced 
algebra  is  better  done  after  a  working  knowledge  of  analytics  and  calculus 
has  been  obtained,  and  work  taught  without  such  interrelation  is  both  more 
difficult  to  master,  and  without  its  full  value.  For  the  purpose  of  engineer- 
ing education,  a  thorough  grounding  in  the  simpler  elements  of  higher  math- 
ematics is  immensely  more  desirable  than  a  perfunctory  acquaintance  with 
rhe  frills  of  mathematical  theory,  such  as  some  of  the  higher  curves  having 
icniarkable  mathematical  properties.  Curves  having  remarkable  properties 
of  engineering  value,  as  for  instance  the  involute,  cycloid,  logarithmic  spiral, 
etc.,  should  be  considered,  as  should  also  the  properties  of  exponential  curves, 
and  the  use  of  logarithmic  cross-section  paper." 

"During  the  college  course  in  algebra  an  attempt  is  generally  made  to 
cover  too  much  ground  rather  than  to  do  the  work  thoroughly.  On  account 
of  algebra  being  often  poorly  taught  in  high  schools  by  the  weakest  teacher, 
like  English  grammar,  many  students  fail  in  this  course." 

"Few  students  are  able  to  transform  a  given  algebraic  formula,  insert 
numerical  constants,  placing  it  in  the  form  most  convenient  for  practical  use, 
for  example,  in  case  of  formulas  relating  to  the  safe  strength  of  materials." 

In  brief,  upon  a  defective  foundation  in  arithmetic  and  algebra,  the  pro- 
fessional mathematician  endeavors  to  erect  a  superstructure  of  the  higher 
mathematics,  which  in  most  cases  never  becomes  a  permanent  part  of  the 
student's  mental  equipment  and  is  therefore  rarely  utilized  and  is  forgotten 
quickly  after  entering  on  the  practice  of  his  profession." 

(n)  Deficiencies. 

5.  In  what  particular  topics  of  algebra  do  students  show  them- 
selves most  deficient  after  the  freshman  year? 

Replies:  abode  f  9 

114          24          17  8          14  9          41 

A  few  replies  mentioned  difficulties  in  more  than  one  topic  of 
algebra.  It  appears  at  once  that  topic  a,  review  of  surds,  exponents, 
quadratics,  etc.,  covers  50%  of  all  the  trouble  recorded.  In  the 
replies  which  gave  specific  details  regarding  a,  surds  were  mentioned 
the  most  frequently ;  and  then  exponents,  quadratics,  imaginaries  and 
manipulation  of  algebraic  symbols  in  about  the  order  given. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  77 

Comments: 

"In  simple  algebraic  reductions,  especially  where  radicals  are  involved." 

"Fundamental  operations ;   such  blunders  as   i/a  -I-  v/&  =  v/a&." 

"In  everything  beyond  quardatics." 

"Solution  of  n  linear  equations  with  n  unknowns  and  of  equations  where 
exponential  functions  are  involved." 

"They  show  a  general  want  of  freedom  owing  to  undigested  loads  of 
information." 

"Inability  to  set  up  equations  applying  to  known  or  observed  facts; 
more  practice  should  be  given  along  that  line." 

"My  students  in  electrical  engineering  are  invariably  deficient  in  their 
algebraic  training  regarding  the  uses  and  powers  of  imaginary  quantities. 
These  factors  and  the  physical  meaning  attached  to  them  are  of  great  im- 
portance in  the  treatment  of  alternating  currents.  I  find,  as  everywhere, 
the  students  are  deficient  in  algebraic  power  through  lack  of  experience 
rather  than  lack  of  extent  of  algebraic  training." 

3)  ANALYTIC  GEOMETRY — (t)  Object. 

6.  Should  the  object  of  analytic  geometry  be  (a)  to  learn  the 
characteristics  and  theory  of  some  class  of  curves,  as  conies,  or,  (b) 
to  learn  a  new  mathematical  language,  using  any  suitable  problems  in 
curves  or  mechanics  as  the  medium  for  the  study  of  this  new  lan- 
guage? 

Replies :  a  b 

56  172 

It  appears  then  that  a  trifle  over  75%  of  the  replies  to  this  ques- 
tion favored  b,  the  learning  of  a  new  language  as  the  object  of  ana- 
lytic geometry. 

Comments : 

"Very  decidely  (b).  Students  generally  fail  to  get  a  definite  grasp  of 
the  relation  between  the  number  of  conditions  a  curve  can  satisfy  and  the 
corresponding  number  of  constants  in  the  equation  of  the  curve.  Some 
exponential  and  logarithmic  curves  should  be  used." 

"Most  decidedly  (&)  is  the  important  element  here,  that  is,  to  learn  a 
new  mathematical  language,  using  any  suitable  problems  in  curves  or  mechan- 
ics as  a  medium  for  the  study  of  this  new  language;  and  to  become  thor- 
oughly familiar  with  it  and  to  acquire  power  in  its  application." 

"Regarding  the  alternative  which  you  offered  between  (a)  and  (fc), 
I  should  be  obliged  to  say  (&),  but  I  would  rather  rephrase  the  matter  my- 
self thus :  I  do  not  believe  that  the  course  in  analytic  geometry  should  be 


78  MATHEMATICS  FOR  ENGINEERS. 

in  any  sense  a  treatment  of  Conic  Sections,  nor  that  it  should  attempt  to 
bring  out  merely  the  geometrical  properties  of  any  certain  class  of  curves ; 
rather  the  whole  object  should  be  to  familiarize  the  student  with  the  notion 
of  representations  of  equations  by  graphical  figures,  and  a  representation  by 
figures  of  functions  divorced  entirely  from  the  equation  idea,  together  with 
as  great  a  familiarity  with  the  simpler  forms  of  curves  as  a  thorough 
treatment  can  give,  so  that  he  retains  at  least  a  rough  knowledge  of  the 
appearance  of  the  curves  in  question;  I  mean  forms  of  the  class: 

ax-}-b,  ax*  -)-  bx  -f-  c,  kx*,  (ax -\-  b)/(cx  -f-  d),  sinx,  cosx^e*,  log  x, 

and  other  simpler  forms,  with  perhaps  some  knowledge  of  the  general 
equation  of  the  second  degree,  certainly  with  a  knowledge  of  such  forms  as 
x*/a*  +  y*/b*  =  i,  etc.;  the  whole  spirit  of  a  class  in  analytic  geometry 
should  be  to  familiarize  the  student  with  this  possibility  of  geometric  helps 
in  algebraic  problems  and  the  representation  of  functions." 

(M)  Importance  of  Topics. 

7.     Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
topics  which  are  fundamental  to  a  course  in  analytic  geometry : 

a.  Algebraic  relations  for  intersection  of  loci,  tangency, 

etc. 

b.  Loci  problems. 

c.  Right  line,  and  circle. 

d.  Conies. 

e.  Higher  curves,  as  cycloids,  etc. 

f.  Coordinates  of  points  in  space  with  application  to  sim- 

ple loci. 


Replies  : 

I. 

II. 

III. 

IV. 

V. 

VI. 

F. 

a 

115 

31 

21 

10 

5 

4 

973 

b 

40 

77 

34 

32 

7 

o 

880 

c 

51 

54 

74 

9 

i 

4 

90S 

d 

ii 

20 

39 

88 

22 

2 

622 

e 

ii 

5 

16 

16 

76 

52 

407 

f 

10 

13 

8 

21 

49 

81 

399 

The  key  for  the  above  table  is  the  same  as  for  question  (2). 
This  question  is  very  closely  related  to  question  (6)  which  should  be 
thought  of  in  connection  with  it.  Nearly  all  the  answers  were  in 
terms  of  a,  b,  c,  etc.,  or  such  as  to  be  directly  translatable  into  these. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  79. 

Comments  : 

"Lacks  ability  to  set  up  problems  in  algebraic  symbols." 
"No  special  emphasis  should  be  given  conies.    They  should  be  regarded 
merely  as  one  class  in  the  discussion  of  curves  in  general  together  with  tri- 
gonometric, logarithmic,  algebraic  and  transcendental  curves." 

(m)  Deficiencies. 

8.  In  what  particular  topic  in  analytic  geometry  do  students 
who  have  completed  the  course  in  that  subject  show  themselves 
most  deficient? 

Replies:  a  b  c  d  e  f 

47  61  8  25  13  20 

In  a  few  instances  this  was  not  answered  in  a,  b,  c,  etc.,  but  in 
statements  which  had  to  be  interpreted  into  that  of  topics  in  question 
(7).  Two  chief  difficulties  were  mentioned  ;  the  inability  of  students 
to  correlate  algebraic  and  geometrical  ideas  and  to  sketch  in  curves 
from  the  inspection  of  their  equation  without  going  into  the  detail 
of  plotting  them. 

Comments  : 

"The  ability  to  sketch  the  locus  from  an  inspection  of  the  equations. 
Determining  the  nature  of  a  locus  from  its  equation  and  interpretation  of 
results." 

"In  expressing  laws  in  algebraic  language  and  deducing  the  general  form 
of  the  locus  from  the  equation." 

"A  general  lack  of  understanding  the  usefulness  of  the  subject  and" 
hence  superficial  knowledge  of  the  fundamentals." 

"In  my  work   they   usually   show   little   knowledge   of    such   curves   as 

y  =  ax*  -f  b  ; 
y  —  ae*  ; 
y  =  asin 


"After  having  had  analytic  geometry  the  students  show  themselves  de- 
ficient in  two  ways  in  their  calculus.  Their  deficiency  is  largely  failure  to 
grasp  the  general  notion  of  plotting  a  function,  and  also  failure  to  appreciate 
the  tangent  problem.  In  their  later  engineering  work  their  deficiency  in 
analytic  geometry  is  largely  on  the  side  of  empirical  curves,  and  obtaining 
of  equations  from  empirical  data.  In  this  they  are  scarcely  to  be  blamed, 
for  this  is  a  new  notation  which  can  at  the  best  be  illustrated  by  lame 
examples  in  their  first  course  in  analytic  geometry." 


So  MATHEMATICS  FOR  ENGINEERS. 

(iv).  Derivatives  and  Integrals  in  Analytic  Geometry. 

p.  Would  it  be  advisable  to  introduce  a  working  knowledge  of 
elementary  differentiation  and  integration  preceding  or  simultan- 
eously with  the  course  in  analytic  geometry? 

Replies :        Yes.   (together) 115 

No 71 

In  addition  to  the  considerable  majority  of  opinion  in  favor 
of  the  early  introduction  of  the  elements  of  the  calculus  (62%  of 
the  total  number  of  the  replies)  the  striking  fact  was  also  brought 
out,  that  all  who  reported  having  tried  this  method  were,  without 
exception,  in  favor  of  it  and  had  found  it  a  success.  On  the  other 
hand  several  who  favored  the  general  proposition  limited  it  to  the 
introduction  of  differentiation  only. 

Comments : 

"No,  haven't  time." 

"Should  begin  this  early  in  algebra." 

"I  'have  combined  both  for  the  last  ten  years." 

"Yes.    I  carry  engineers  thru  analytics  and  calculus  simultaneously." 

"I  think  it  highly  advisable  to  treat  the  tangent  problems  by  this  method 
and  to  give  the  student  a  working  knowledge  of  the  simpler  processes  of 
differentiation  along  with  their  analytic  geometry." 

4)  AIDS  AND  REMEDIES  FOR  ABOVE. 

10.     What  is  the  remedy  for  (j),  (5),  and  (8)? 

The  replies  to  this  question  cannot  be  exhibited  in  the  tabular 
form  previously  used.  The  remedy  given  most  frequently  was  to 
apply  the  theory  to  practical  problems.  Two  reasons  were  found 
for  this;  first  that  the  greatest  difficulty  met  by  students  later  on 
was  in  the  application  of  principles  which  had  been  learned,  and 
second,  that  this  would  create  interest  and  realization  of  the  im- 
portance of  the  topics  studied.  Another  remedy  frequently  proposed 
was  "more  drill,  more  drill"  and  in  direct  connection  with  this  the 
plea  for  frequent  reviews. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  81 

Comments : 

"Review  constantly." 

"Special  short  course  sophomore  year." 

"Less  extension  and  more  intension  in  study  and  drill." 

"Make  necessary  topics  more  important,  other  less  so." 

"Give  simple  problems  illustrating  laws  of  mechanics  and  physics." 

"Smaller  number  of  students;  copious  problems  with  engineering  appli- 
cation ;  instructors  who  know  needs  of  engineer." 

"No  one  tilling.  Students  would  do  well  to  think  more  and  hurry  less. 
Perhaps  it  would  be  well  if  writers  spent  more  time  in  pointing  out  errors 
that  have  been  made  and  are  to  be  guarded  against." 

"To  extend  the  time  devoted  to  the  subject  so  as  to  admit  of  a  more 
extensive  drill,  and  to  draw  upon  engineering  practice  for  concrete  problems 
which  will  arouse  the  student's  interest  and  hold  his  attention." 

"Greater  accuracy  should  characterize  the  high  school  and  grades.  The 
habit  of  accuracy  can  not  be  developed  in  a  day  or  in  a  year." 

"Emphasize  mathematics  as  a  tool  and  have  it  thoroughly  mastered 
from  that  standpoint." 

"More  drill ;  selection  of  problems  they  will  meet  in  second  year  and  lay 
stress  upon  that  fact." 

"Cut  out  at  least  one-third  of  the  theoretical  stuff  and  use  the  time 
given  to  it  in  practical  arithmetic  and  graphical  problems  in  engineering." 

"Working  many  problems  taken  as  far  as  possible  from  practical  appli- 
cations either  real  or  possible.  The  practical  application  awakens  interest, 
and  interest  overcomes  many  difficulties.  We  should  in  all  cases  show  the 
relation  as  noted  in  your  section  (6b).  The  problems  should  be  carefully 
corrected,  not  merely  marked  or  other  methods  of  solution  sihown,  but  the 
student's  fallacies  or  errors  pointed  out.  There  should  be  frequent  quizzes 
or  drills.  I  prefer  to  devote  the  first  half  of  every  period  to  drill  and  the 
last  half  to  explanations  of  next  lesson  and  clearing  up  any  difficulties  in 
quiz  just  given.  The  quizzes  should  overlap  somewhat;  i.e.  should  include 
work  done  in  the  last  quiz  or  two,  or  there  should  be  a  general  quiz  every 
5th  period." 

"Too  much  stuffing  and  learning  by  rote." 

"Early  training  to  be  analogous, to  that  in  English  and  German." 

"Redistribution  of  emphasis  in  teaching." 

"A  logical  course  in  mathematics  extending  three  years,  not  algebra, 
trigonometry,  etc.,  as  usually  taught." 

"Vitalizing  mathematical  work  by  giving  more  time  to  its  concrete 
application  in  physics,  chemistry  and  engineering  and  less  to  mathematical 
training  seldom  of  use  in  practice." 

"I  have  noticed  no  predominating  deficiency.  >The  general  deficiency 
is  failure  to  recall  and  apply  the  knowledge  attained  earlier.  To  forestall 


82  M  A  THEM  A  TICS  FOR  ENGINEERS. 

this  as  far  as  posible  it  would  seem  advisable  that  the  Freshman  instructor 
make  clear  the  purpose  of  the  subject.  Make  the  few  fundamental  principles 
and  methods  stand  out  boldly.  Aim  to  cultivate  the  mathematical  type  of 
thought.  Show  how  the  few  basic  methods  of  each  subject  act  in  slightly 
different  varying  ways  to  solve  many  important  problems  and  give  consider- 
able practice  in  actually  doing  this." 

"The  subjects  of  algebra,  trigonometry  and  analytics  should  be  less, 
rigorously  separated  into  water-tight  compartments.  The  solution  of  simple 
plane  triangles  should  be  brought  in  with  plane  geometry,  using  only 
natural  functions  of,  say,  acute  angles.  Considerable  trigonometry  can  be 
brought  out  in  analytic  geometry  and  still  more  in  calculus.  Algebra  and 
trigonometry  can  be  united  in  work  in  series  to  get  roots  of  imaginary 
complex  quantities  by  DeMoivre's  Theorem.  Whenever  a  new  method  has 
been  developed,  the  teachers  should  point  out  and  emphasise  the  practical 
problems  and  examinations  which  do  not  presume  that  a  grade  in  a  subject 
makes  continued  knowledge  in  that  subject  necessary." 

"The  remedy  for  questions  (j),  (5),  and  (5)  in  my  judgment  is 
presistent  practice  required  of  the  students  in  application  of  principle  to 
practical  problems.  The  use  in  explanation  by  the  instructor  of  the  simplest 
non-technical  language  and  abundant  reference  to  the  homeliest  illustrations 
at  his  command.  I  believe  tihat  the  crucial  difficulty  in  the  case  of  the 
average  student  of  mathematics  is  his  fear  of  the  subject  born  largely  of  a 
too  generous  use  by  instructors  of  the  technical  and  unfamiliar  language,  and 
the  failure  to  use  wherever  possible,  illustrations  drawn  from  things  with 
which  the  student  is  commonly  familiar.  We  also  correlate  as  thoroughly 
as  may  be,  the  subjects  which  are  allied  to  mathematics  with  tihe  mathematical 
instruction  itself;  as  for  instance,  applied  mechanics,  physics  and  applied 
electricity." 

"For  question   (8)  the  tracing  of  all  curves  met  with  further  on." 

"Some  algebra  should  be  taught  every  year  in  the  high  school." 

"Make  trigonometry  more  simple.  Use  more  coordinate  and  graphic 
methods." 

"Connect  analytic  geometry  with  calculus." 

"Drill  on  (a)  in  each  witih  special  attention  to  clear  statements  and  full 
complete  reasoning." 

"For  questions  (j)  and  (8)  give  greater  emphasis  on  algebraic  nota- 
tion as  shorthand  translation  for  English  into  algebra  and  vice  versa." 

"At  the  beginning  of  analytic  geometry  a  short  and  thoro  drill  on 
the  kind  of  algebraic  process  and  trigonometric  relations  to  be  used  is 
advised." 

"A  similar  drill  just  before  starting  calculus  on  some  of  the  most 
needed  algebraic,  trigonometric  and  analytic  relations  would  largely  remedy 
the  difficulty.  For  example,  a  sharp  drill  on  a  few  of  the  most  fundamental 
relations  of  trigonometry  would  help  in  differentiation  and  integration  of 
trigonometric  forms." 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  83 

V.    MODIFICATIONS  IN  MATHEMATICS  FOR 
ENGINEERS. 

i)  ENGINEERING  PROBLEMS — (t)  Advisability. 

ii.  Is  it  advisable  to  treat  problems  met  in  actual  engineering 
work? 

Replies :        Yes  190 

No 20 

The  answers  to  this  question  while  overwhelmingly  in  the 
affirmative — a  little  over  90% — contained  many  warnings  against 
overdoing  and  the  use  of  too  technical  material.  They  also  strongly 
condemned  the  use  of  such  problems  by  instructors  ignorant  of  the 
physical  meaning  of  the  quantities  involved.  The  principal  objec- 
tions were  the  lack  of  time  and  the  lack  of  such  problems. 

Comments : 

"Very  seldom.    We  want  principles  and  not  applications." 
"No,  because  such  problems  have  too  many  unknown  quantities  in  them." 
"No,  k  is  better  to  use  ideal  problems  of  physics,  chemistry,  etc.'' 
"Yes,  but  there  are  objections : — Not  sufficient  time.    No  sucih  problems 
perhaps  at  hand.     Difficult  to  get  problems  of  requisite  simplicity  and  near 
enough  to  actual  engineering  problems  to  be  of  any  value." 
"Yes,  without  duplicating  teaching." 

"Yes,  adds  interest  but  they  must  necessarily  be  elementary." 
"Simple  problems.    Do  not  make  the  common  mistake  of  using  appli- 
cations which  are  Greek  to  the  students." 

"Yes,  thorough  drill  in  problems  met  with  in  land  surveying." 
"Regarding  the  problems  from  actual  engineering  work,  I  would  say 
that  I  should  favor  very  strongly  th«  introduction  of  such  problems,  pro- 
vided they  are  so  selected  as  to  be  within  the  reach  of  the  student's  present 
knowledge.  They  should  not  be  absurd,  as  are  some  problem's;  i.e.,  they 
should  correspond  to  some  real  facts,  and  they  should  not  be  fake  practical 
problems.  They  should  not  be  a  statement  of  such  a  somebody's  formula, 
where  the  various  letters  mean  certain  unheard-of  things.  These  restric- 
tions limit  the  possible  material  enormously,  but  I  am  convinced  that  they 
should  be  insisted  upon." 

"Yes,  to  bring  teachers  of  mathematics  into  close  touch  with  teachers 
of  engineering." 


84  MATHEMATICS  FOR  ENGINEERS. 

(«').     Nature  of  Such  Problems. 

12.  If  so,  to  what  extent  cuid  under  which  topics  of  (2),  (4) 
and  (7)? 

Replies : 
For  (2} 

a.  Derivation  and  manipulation  of  formulae 1 1 

b.  Solution  of  triangles  by  use  of  natural  function 19 

c.  Use  of  logarithms  in  solution  of  triangles  and  other  com- 
putations  35 

d.  Problems  applying  the  solution  of  triangles 39 

For  (4) 

a.  Review  of  surds,  exponents,  quadratics,  etc 20 

b.  Series   19 

c.  Binomial  Theorem  13 

d.  Permutations  and  combinations 4 

e.  Graphs    26 

/.  Determinants I 

g.  Theory  of  equations 12 

For  (7) 

a.  Algebraic  relations  for  intersection  of  loci,   tongency, 
etc 18 

b.  Loci  problems 28 

c.  Right  line,  and  circle 23 

d.  Conies    17 

e.  Higher  curves,  as  cycloids  etc 7 

/.     Coordinates  of  points  in  space  with  application  to  simple 

loci    7 

The  answers  which  referred  to  explicit  topics  in  questions  (2), 
(4),  and  (/)  are  embodied  in  the  above  table.  This  means  that 
thirty-nine  replies  suggested  d  for  question  (2).  d  was  not  the 
only  suggestion  in  these  thirty-nine  replies,  however,  as  several 
contained  two  or  three.  While  the  number  of  answers  received  is 
not  so  very  large  the  opinions  expressed  are  quite  uniform,  as  will 
be  seen  by  referring  to  the  above  table. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  85 

Comments : 

"To  limited  extent  only  under  any  topic  needed  in  solution." 

"Only  a  few  problems.  The  tendency  in  modern  text-books,  is  to  treat 
-too  many  problems  of  the  kind." 

"In  all,  mainly  in  (2)  and  (7)." 

"Graphic  solutions  wherever  possible." 

"Follow   a   principle   with   an    illustration." 

"Whenever  possible.  They  should  require  little  technical  knowledge, 
It  is  easy  to  over  do  this  thing." 

"Problems  applying  solution  of  triangles,  series,  graphs,  conies  and  higher 
.curves." 

"Paths  for  (7).  Polygon  of  forces  for  (4)  and  (<?),  (2)0  resolution  of 
forces,  bridge  structures,  etc.,  (7)0  gear  teeth,  (/)/  intersections  of  solids, 
with  models." 

"(2)b,  c,  d  surveying  and  railroad  curves." 

"To  awaken  interest  and  show  relation  between  abstract  and  applied 
mathematics: — (2)  b,  d;  (4)  c,  f,  g,  d;  (7).  Call  attention  to  their  appre- 
ciation rather  than  problems." 

"The  engineering  problems  should  come  as  exercises  of  the  application  of 
mathematical  principles,  rather  than  as  a  means  of  teaching  those  principles. 
These  illustrative  problems  should  be  introduced  in  connection  with  every 
topic  possible.  I  believe  that  the  reason  so  much  of  the  mathematics  studied 
by  the  engineering  students  falls  into  disuse  and  becomes  forgotten,  is  that 
the  average  student  has  to  be  shown  how  to  use  practically  what  has  been 
studied  theoretically,  and  if  not  shown,  will  never  get  the  practical  use  out 
of  what  he  has  learned.  The  mind  seems  to  be  so  occupied  in  grasping  the 
mathematics  involved,  that  the  application  is  lost  unless  specifically  illustrated." 

2)  ADVISABILITY  OF  A  COURSE  IN  COMPUTATION. 
/j.     Is  a  short  course  in  computation  advisable  f 

Replies :        Yes    124 

No   49 

Several  who  favored  such  a  course  added  either  qualifying 
statement,  "if  there  is  time  for  it,"  or  "if  there  is  need  for  it." 
Many  who  opposed  such  a  course  were  really  in  favor  of  the  plan 
except  for  time  it  would  require.  In  one  quotation  under  question 
20  an  important  point  is  brought  out;  namely,  that  while  students 
are  continually  admonished  to  be  accurate,  they  are,  as  a  rule,  told 
very  little  about  the  degree  of  accuracy  required  by  the  various 
problems  encountered  and  given  next  to  no  practice  in  the  same. 


86  MATHEMATICS  FOR  ENGINEERS. 

Comments: 

"Good,  but  no  time  for  it." 

"Should  be  emphasized  in  all  courses  but  not  given  as  a  separate  course." 

"A  large  number  of  computation  problems  should  be  given  in  connection 

with  the  different  courses.     I  would  not  have  separate  computation  courses." 
"Review." 

"By  all  means,  a  short  course,  or  a  course  to  be  made  as  long  as  possible." 
"Yes,  particularly  if   supplemented  by   graphical   methods;   i.e.,   use  of 

slide  rules." 

3)  SEGREGATION  OF  ENGINEERS  IN  MATHEMATICS  —  (i)  Advisability* 

14.     Should  freshmen  engineers  be  taught  their  mathematics 
in  separate  classes  or  together  u*ith  those  taking  other  courses? 

Replies  :        Separate    .............................  136 

Together  .............................   55 


of  the  opinions  expressed  were  thus  in  favor  of  the  special 
sections  for  freshmen  students  in  engineering.  A  few  who  voted 
"together"  did  so  because  the  practical  training  needed  by  engineers 
would  also  be  the  best  for  the  literary  and  general  science  students. 
The  reasons  given  are  taken  up  under  the  next  question. 

(ii)  Reasons. 
15.    Why? 

In  the  following  quotations  those  under  (a)  state  reasons  why 
separate  classes  are  unnecessary,  those  under  (b)  why  they  are 
necessary,  and  those  under  (c)  why  the  literary  and  general  science 
students  should  take  their  mathematics  in  classes  organized  for 
the  engineering  students. 

Comments  : 

(a)  "There  is  no  engineering  mathematics." 

"Good  mathematics  for  one  is  good  mathematics  for  the  other." 
"Aids  in  cultivating  wider  social  sympathy." 
"In  order  to  not  lose  its  culture  value  apart  from  application." 
"Because   mathematics   should   be   studied   as   a   science   rather   than   as 
a  mere  working  tool  for  engineers  to  be  applied  mechanically." 

(b)  "So  as  to  emphasise  engineering  application." 

"Mathematics  for  engineers  has  two  objects:     tftie  development  of  the 
student  and  the  practical  value." 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  87 

"The  engineers  aim  at  facility  in  computation  and  the  use  of  formulas, 
while  the  art  students  go  more  into  foundation  and  theory." 

"Because  the  men  in  other  courses  will  not  probably  appreciate  the 
spirit  in  which  the  subject  should  be  attacked.  Not  because  the  method  or 
course  should  be  different." 

"Mathematical  rigor  need  not  be  insisted  upon.  Either  to  select  illus- 
trative material  which  will  apply  to  whole  class." 

"Realization  of  the  greater  certainty  of  the  practical  use  made  of  their 
mathematics  will  probably  result  in  the  average  engineering  student  ac- 
complishing more  than  will  be  accomplished  by  the  non-engineering  student, 
if  the  former  is  unhampered  by  the  company  of  the  latter." 

"Because  the  instruction  for  the  freshman  is  the  most  important  in- 
struction, especially  in  mathematics  during  the  four  years'  course.  The 
freshmen  require  a  very  specific  and  individual  treatment  as  far  as  possible. 
It  seems  so  fundamental  that  they  should  be  taught  right  ways  of  thinking 
as  well  as  right  mathematical  facts,  that  they  ought  to  be  a  special  and  entirely 
separate  problem." 

(c)  "What  is  good  for  the  engineer  will  be  good  for  others,  even  the 
practical  problems." 

''The  emphasis  of  the  practical  does  not  detract  from  the  educational 
nature  of  the  subject." 

4)  SPECIAL  TEXTS. 

1 6.    Are  special  texts  in  mathematics  for  engineering  students 
advisable? 

Replies :        Yes    93 

No  89 

Nothing  of  importance  was  brought  out  by  this  question  other 
than  the  results  given  above  and  the  quotations  which  are  appended. 

Comments : 

"They  are  not  as  a  rule  satisfactory." 

"Rather  special  teachers." 

"Only  if  it  is  written  more  clearly." 

"Not  necessary.  Supplementary  material  should  be  chosen  with  re- 
ference to  students." 

"Such  texts   may  save  the  students   valuable  time." 

"No,  all  texts  should  be  given  a  turn  toward  engineering." 

"Perhaps  for  analytics  and  calculus  but  not  for  algebra." 

"Only  for  analytic  geometry  where  there  is  room  for  a  real  working 
text." 

"Yes,  if  the  teacher  can  not  supply  what  is  lacking  in  the  theoretical 
text-book." 

"Yes  and  no.    Theoretically  it  would,  but  I  know  of  no  such  texts." 


88  MATHEMATICS  FOR  ENGINEERS. 

VI.    PEDAGOGIC  QUESTIONS. 

1)  QUIZZES. 

77.    How  often  should  written  quizzes  be  given? 

Of  the  181  formal  answers  received  32  thought  quizzes  should 
be  given  monthly,  TO  every  there  weeks,  55  every  two  weeks,  22 
weekly,  25  frequently,  I  daily,  I  seldom,  and  35  at  the  option  of  the 
instructor.  The  general  expression  was,  of  course,  that  no  fixed 
rule  could  be  given. 

Comments : 

"As  often  as  instructor  can  stand  them.  Should  be  left  entirely  to- 
him." 

"Frequently,  with  enough  time  given  to  allow  the  students  to  think  out 
the  problems." 

"In  my  work  I  find  it  so  full  of  interest  to  the  pupils  and  the  time  at 
my  disposal  so  limited  th&t  I  dread  to  lose  good  time  for  the  purpose  of 
a  quiz.  Every  quiz  means  a  lot  of  good  information  lost  to  the  class." 

2)  MODES  OF  CONDUCTING  CLASSES. 

18.  Number  i,  2,  3,  etc.,  the  order  of  usefulness  of  the  follow- 
ing modes  of  conducting  classes,  suggesting  any  further  modes  or 
combinations  of  these: 

a.  Assignment  of  work  from  text  without  any  previous  ex- 

planation. 

b.  Assignment  of  zvork  from  text  ivith  previous  explanation. 

c.  New  work  explained,  the  student  taking  notes — rvith  or  with- 

out the  use  of  a  text.     Written  or  oral  quizzes  at  intervals. 

d.  Students  led  to  discover  new  principles  by  suggestions  and 

quizzes  on  the  old. 

e.  Each  student  working  independently,  going  as  fast  as  he 

is  able.     This  may  apply  to  the  entire  course,  to  each  topic 
or  to  each  day's  work. 

f.  First  part  of  the  hour  de^foted  to  the  class  as  a  whole;  second 

Part  given  to  individual  work. 

g.  Subject  treated  from  the  laboratory  standpoint;  most  of  the 

work  being  done  in  the  class  room. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  89 

Replies:  I.          II.      III.        IV.      V.      VI.      VII      F. 

a  62        18        20        17        ii         12          7        774 

b 

c 

d 


78 

40 

12 

9 

8 

4 

5 

916 

30 

24 

29 

18 

12 

8 

9 

632 

i6 

23 

29 

17 

2O 

7 

6 

543 

7 

7 

II 

ii 

13 

22 

28 

271 

40 

37 

31 

30 

II 

4 

2 

820 

ii 

19 

8 

16 

17 

21 

2O 

39i 

Several  considered  any  definite  answer  to  this  question  impos- 
sible. The  objections  were  that  (i)  these  modes  are  so  intertwined 
with  each  other  as  to  be  nearly  inseparable,  thus  making  it  often  nec- 
essary to  use  several  in  one  recitation,  (2)  that  different  treatment 
was  necesary  for  different  classes  and  (3)  for  different  kinds  of 
work.  The  result  of  the  letters  was  that  certain  definite  combina- 
tions are  quite  feasible,  and  often  adhered  to  quite  rigorously  by 
some  instructors  for  not  only  a  single  recitation  but  are  used  as  a 
general  plan  of  conducting  classes.  Some  of  those  mentioned  were 
"a,  b  and  d,"  "a  and  /."  In  a  few  cases  one  or  two  double  periods 
per  week  were  given  to  work  conducted  along  the  line  explained  in  g. 
In  a  few  of  the  letters  e  and  g  were  considered  faddish  and  worth- 
less. On  the  other  hand  quite  a  number  of  instructors  who  could 
not  use  these  modes  because  of  too  large  classes  nevertheless  ranked 
them  first  and  considered  them  to  be  "ideal  for  small  classes."  No 
particular  objections  and  no  special  variations  in  modes  depending 
upon  subject  matter  were  expressed. 

Comments  : 

"I  depend  on  circumstances." 

"g  and  e  desirable  but  generally  not  practicable." 

"c  and  d  can  easily  be  overdone,  also  g." 

"I  generally  combine  b  and  /.  ' 

"o  and  /.  Engineering  students  seem  to  do  best  if  they  put  problems 
and  proofs  on  the  board  and  explain  daily." 

"e  is  ideal  if  one  instructor  can  be  assigned  to  each  three  or  five 
students." 

"I  would  favor  a  and  /.  It  is  a  waste  of  time  to  try  to  explain  or  lecture 
to  the  class  until  they  have  spent  considerable  time  in  the  preparation." 

"We  are  trying  some  experiments  which  might  be  classed  under  b,  g,  and  / 
but  have  not  yet  gone  far  enough  to  draw  any  inference.  To  gain  good 
results  I  believe  it  absolutely  necessary  that  the  instructor  should  be  able 


90 


MATHEMATICS  FOR  ENGINEERS. 


to  give  individual  attention  to  the  students  in  his  charge,  and  this  not  only 
once  a  month  but  every  hour  they  come  to  his  class  room  or  laboratory." 

"The  laboratory,  wherever  possible,  is  a  most  effective  way  of  clinching 
points  made  in  the  recitation  room  and  of  clearing  up  points  that  are 
vague.  A  few  hours'  work  in  a  laboratory  having  proper  facilities  will  cause  a 
student  to  better  grasp  the  real  meaning  of  'moment  of  inertia*  than  many 
hours  of  class-room  work.  Laboratory  and  class-room  work  should  always 
be  worked  conjointly." 

"My  own  mode  is  a  combination  of  g  and  d.  I  am  one  of  probably 
very  few  teachers  who  can  say,  I  have  taught  school  twelve  years  and  have 
yet  to  conduct  my  first  recitation.  I  am  not  sure  my  method  could  be 
applied  to  the  teaching  of  mathematics,  but  I  surely  hope  such  is  the  case. 
I  would  rate  e  first  if  it  were  practically  possible.  We  have  faithfully  tried 
it — failure." 


VII.    GENERAL  NEEDS  AND  DEFICIENCIES. 

Besides  the  needs  and  difficulties  pertaining  to  any  particular 
topic  as  considered  in  question  2  to  10  there  are  those  of  a  more 
general  nature  as  the  power  to  think  mathematically,  or  ability  to 
analyze  problems.  These  are  considered  in  the  next  four  questions. 

i)  GENERAL  MATHEMATICAL  ABILITY — (t)  Need. 

ip.  Number  i,  2,  3,  etc.,  in  order  of  importance  the  following 
pertaining  to  the  qualifications  of  the  engineering  student  who  has 
completed  the  course  in  freshman  mathematics: 

a.  Skill  and  accuracy  in  computations. 

b.  Analysis  of  problems. 

c.  Interpretations  of  results  in  solution  of  problems. 

d.  Knowledge  and  use  of  equations. 

e.  Representation  of  physical  laws  by  means  of  graphs. 

Replies:  L  II.  III.  IV.  V.  F 

a  93  41  37  26  12  804 

b  122  52  17  6  9  880 

c  41  66  63  25  7  715 

d  23  35  42  56  32  525 

e  13  19  30  50  75  406 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  91 

Of  the  general  mathematical  qualifications  enumerated  in  the 
question,  the  tabulated  results  show  conclusively  that  a,  b  and  c 
are  considered  the  chief  ones,  with  special  emphasis  on  b,  the 
analysis  of  problems.  It  is  in  this  latter  that  the  most  difficulties 
are  also  found,  as  will  be  seen  under  20.  The  replies  indicate  that 
the  trouble  students  have  with  the  application  of  mathematical  theory 
in  problems  is  a  case  of  knowing  "how  to  do"  while  not  knowing 
"what  to  do." 

Comments : 

"a  and  b  are  useless  without  the  others.  They  are  of  prime  import- 
ance." 

"It  is  my  opinion  that  students  invariably  show  themselves  most  defi- 
cient in  analysis  of  problems,  that  is,  in  transferring  the  plain  English 
statements  of  a  problem  into  mathematical  form,  in  shape  to  be  handled  by 
the  mathematical  processes  with  which  they  are  often  familiar." 

"A  practicing  engineer  seldom  goes  back  to  fundamentals,  but  works 
largely  with  standard  tables  and  formulas,  and  his  mathematical  training  is 
only  of  advantage  to  him  to  enable  him  to  use  these  readily  and  intelligently, 
and  to  understand  the  history  and  origin  of  these  so  that  he  can  apply  them 
without  fear  of  misusing  or  abusing  them  as  non-tedhnical  engineers  are 
likely  tto  do.  If  an  engineer  has  the  time  and  opportunity  to  go  into  research 
work,  he  may  then  call  into  play  his  theoretical  mathematics.  It  is  well  to 
train  young  engineers  for  that  possibility,  but  the  great  majority  have  little 
opportunity  for  it.  The  mechanical  branches  algebra,  analytic  geometry,  cal- 
culus and  descriptive  geometry  and  graphics,  are,  however,  of  the  utmost 
value  in  the  way  I  have  indicated,  and  a  great  deal  of  drill  work  should  be 
given  in  these  branches  and  plenty  of  problems  of  as  original  a  character 
as  possible.  It  is  of  more  importance  to  cultivate  in  the  student  the  power 
and  ability  to  attack  a  problem  directly  and  intelligently  and  to  solve  it  by 
short  and  sensible  means  than  to  learn  many  theories." 

"I  find  difficulty  in  arranging  the  order,  c  is  the  whole  thing  from  my 
standpoint,  if  one  means  by  that-  the  understanding  of  the  theory  in  its 
application  to  examples  under  that  theory.  I  mean  the  grasp  by  the  student 
of  the  fundamental  meaning  of  the  subject  under  consideration.  That  is 
the  large  thing.  If  the  student  does  that  he  can  be  pretty  well  trusted  to 
get  those  minor  details,  the  important  formulas,  the  methods  of  solutions  of 
problems  and  all  that,  a  enters  very  largely  in  trigonometry  otherwise 
-would  not  be  of  vital  importance,  though  accuracy  should  always  be  given  a 
prominent  place.  In  comparison  with  my  remark  under  c  the  topics  under 
a,  b,  d  and  e  are  of  absolutely  minor  importance ;  still  all  four  of  these  are 
quite  important  and  ought  to  be  insisted  upon  rather  largely." 


92  MATHEMATICS  FOR  ENGINEERS. 

(it)  Deficiencies. 

20.    In  which  of  (/p)  do  students  who  hare  completed  their 
freshman  year  show  the  greatest  deficiency? 

Replies:  a  b  c  d  e 

68  93  70  19  ii 

Comments: 

"They  can  not  apply  principles  to  practical  work." 

"Usually  students  show  a  greater  weakness  in  analysis  than  in  many  other 
topics  and  as  a  result,  the  accuracy,  the  interpretation,  etc.,  are  likewise 
deficient." 

"As  an  example  of  trouble  in  mathematics  the  following  may  be  of 
interest.  I  gave  this  problem  to  40  senior  engineers :  A  steam  boiler  when 
tested  shows  an  efficiency  of  65%.  After  providing  it  with  an  economizer 
it  shows  85%.  What  is  the  percentage  of  saving  of  fuel?  Half  of  the  class 

Of        far 

gave    the    answer  in  this   fashion      •> '    5    .=  30.75%    and    the    other    half 

05 

Q-       /:  f 

-2- — 2.       —  23.53%.    Only  one  of  the  40  gave  a  logical  statement  to  show 
°5 

why  the  formula  he  used  was  right." 

"As  regards  a,  one  great  deficiency  in  all  students  is  a  lack  of  knowledge 
in  regard  to  what  degree  of  accuracy  is  called  for  in  any  calculation.  They 
have  heard  a  great  deal  too  much  about  accuracy,  and  not  enough  advice  has 
been  given  as  to  what  is  good  enough  for  the  given  problem,  or  how  to 
determine  the  required  degree  of  accuracy." 

''The  weak  point  in  the  work  of  engineering  students  in  the  upper 
class  courses  is  usually  their  inability  to  apply  mathematical  analysis  to 
practical  problems,  and  to  formulate  mathematical  expressions  for  physical 
facts.  To  my  mind,  the  remedy  for  this  difficulty  lies  in  emphasizing  the 
physical  rather  than  the  philosophical  aspect  of  mathematics,  and  of  attach- 
ing to  every  mathematical  expression  and  operation  in  a  problem,  a  definite, 
clear-cut  physical  meaning.  The  teaching  of  physics  and  mechanics  as 
parallel  courses  with  the  calculus  is  a  very  useful  method  of  accomplishing 
this  result.  The  constant  use  of  easy  practical  problems,  in  which  the  student 
can  thoroly  grasp  the  physical  theory  of  the  problem,  is  very  useful  in  every 
kind  of  mathematical  teaching." 

(Hi)  Remedy. 

<?/.     W hat  is  the  remedy  f 

The  remedies  proposed  for  difficulties  met  under  a,  may  be 
briefly  summarized  as  follows:    alwavs  to  demand  accurate  results 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  9$ 

in  any  computation,  to  regard  checks  upon  all  results  as  a  part 
of  the  solution  of  the  problem,  and  to  urge  upon  all  teachers  of 
secondary  mathematics  the  necessity  of  insisting  upon  checking. 

The  suggestions  for  b  and  c  can  not  be  presented  in  any  such 
definite  form  as  for  a.  Insistance  upon  analysis  of  problems  from 
the  grades  up,  the  following  of  every  theory  learned  with  illustra- 
tive applications^  and  the  elimination  of  work  by  formulas  and 
rote  as  far  as  possible,  are  strong  points.  Practically  no  suggestion 
for  improvement  was  offered  regarding  d,  and  nothing  was  said 
regarding  e. 

Comments : 

"More  concrete  problems  to  awaken  interest." 
"Better  and  more  attractive  texts  and  more  efficient  instructors." 
"Make  students  do  the  work  rather  than  instructors." 
"The  remedy  is  to  be  found  in  further  training  not  in  the  freshman 
year." 

"Application   and   interpretation   of  theory;   many  problems." 
"More  drill  and  better  training  in  English." 

"Begin  with  public  school  work  and  drill,   drill,   drill.     Pretty  hard  in 
one  freshman  year  to  undo  years  of  slovenly  habits  and  lack  of  thought." 
"Before  working  a  problem  before  the  class  give  a  thorough  analysis 
of  the  principles  involved." 

"Frequent  exercises  in  analysis  by  use  of  problems  slightly  different. 
Avoid  monotony  and  sameness.  Make  the  men  defend  each  position  and 
make  good  every  assertion." 

"Learn  to  read  equations  in  analytic  geometry." 

"Most  decidedly  more  training  in  the  use  of  what  they  are  getting  in 
mathematics." 

"Constant  comparison  and  reference  to  actual  practical  work.  Let  stu- 
dent remain  out  of  school  for  a  year  and  devote  himself  to  some  real 
practical  work  under  a  boss." 

"Require  a  rigid  checking  system." 

"Special  work  in  computation  in  connection  with  course  in  trigonometry." 
"More  drill  in  fundamentals  and  add  course   (13)." 
"Require  accuracy  from  beginning  of  the  mathematical  course." 
"Insist  on  accuracy  as  of  equal  importance  with  other  items." 
"Have  students  work  many  problems  and  never  use  a  key  or  answers." 
"Give  problems  for  home  solution  along  the  line  needed." 
"I    realize   that   it   is   practically   impossible   in   a   four   year   course   to 
teach   everything,   but   for   the   engineering  graduate  who   expects  to   make 
a   financial    success    in   his   after   life    and   not  simply    remain   an    entirely 
theoretical  man  and  therefore  always  on  a  salary,  an  employee  of  someone 


I. 

II. 

III. 

F. 

159 

20 

25 

642 

34 

67 

"3 

349 

59 

108 

37 

420 

.94  MATHEMATICS  FOR  ENGINEERS. 

else,  it  has  always  seemed  to  me  that  if  the  minds  of  'the  students  could 
be  turned  a  little  bit  more  toward  the  practical  end  of  the  work,  in  addition 
to  their  theoretical  ideas,  it  would  in  the  end  be  greatly  to  their  advantage." 

2)  RELATIVE  IMPORTANCE  OF  TOPICS. 

22.  Designate  by  i,  2,  j,  etc.,  the  order  of  importance  for  any 
mathematical  topic  of: 

a.  Mastery  of  theoretical  matter  involved. 

b.  System  and  neatness. 

c.  Accuracy  in  computed  results. 

Replies : 
a 
b 
c 

It  is  clear  that  the  majority  have  placed  the  above  in  the  order 
a,  c,  b.  Where  these  were  said  to  be  of  equal  importance  all  were 
given  the  rank  I. 

Comments : 

"Without  accuracy  all  are  worthless." 
"By  no  means  should  there  be  any  differentiation  here." 
"They  are  so  interdependent  it  is  difficult  to  give  any  answer.     System 
and  neatness  are  necessary  to  insure  accuracy." 


VIII.    ENGINEERING  vs.  A  GENERAL  EDUCATION. 

23.  Is  it  true,  as  has  been  said,  that  engineering  students  apply 
themselves  better  than  those  taking  a  literary  or  scientific  course? 

Replies :        Yes    125 

No  45 

Many  said  that  as  they  had  not  had  experience  with  both 
classes  of  students  they  considered  themselves  as  incompetent  of 
expressing  any  opinion  on  the  question.  This  has  reduced  the  numr 
ber  of  replies  somewhat. 


CURRENT  THOUGHTS  ON  VITAL  QUESTIONS.  95 

Comments : 

"At  some  institutions  it  is,  at  others  not;  in  general  it  is  true  in  the 
east  but  not  in  the  Mississippi  valley  and  the  west." 

"Yes,  in  mathematics ;  much  better  than  the  average  student  in  literary 
or  scientific  courses,  but  not  nearly  as  well  as  the  best." 

"I  think  it  is  true  that  the  engineering  students  apply  themselves  better 
than  those  taking  literary  or  scientific  courses  but  am  not  sure  that  their 
course  is  the  more  thorough.  There  are  scientific  students  who  do  their 
work  much  more  thorough  than  the  average  engineering  students,  but  the 
average  engineering  student  has  a  definite  idea  of  what  Hie  wants  and  he 
seeks  those  practical  results." 

"I  believe  that  the  training  derived  from  an  appropriately  taught  thorough 
engineering  course  is  fully  the  equivalent  of  that  obtained  from  a  general 
science  or  literary  course.  In  general,  the  grade  of  teaching  done  in  engi- 
neering schools  is  inferior  to  that  done  in  classical  schools,  and  the  student 
is  not  carried  to  that  point  in  his  work  where  the  training  is  of  its  highest 
developmental  value.  For  instance,  emperical  machine  design,  taught  without 
the  foundation  of  applied  mechanics,  is  utterly  without  developmental  value, 
rational  machine  design  has  great  value  along  such  lines.  When  taught 
for  the  purpose  of  developing  the  power  of  analysis  in  a  certani  branch  of 
useful  knowledge,  engineering  courses  are  good.  When  taught  tor  the 
sole  purpose  of  turning  out  men  capable  of  accomplishing  certain  useful 
tasks  in  an  orthodox  and  correct  manner,  engineering  courses  are  of  ab- 
solutely no  developmental  value.  The  first  method  of  engineering  education 
tends  to  turn  out  keen,  clear,  logical  minds,  capable  of  applying  their  power 
to  any  task,  whether  it  is  a  problem  in  thermodynamics,  in  machine  shop 
practice,  or  in  public  finance.  The  second  method  tends  to  turn  out  men  of 
immediately  available  but  limited1  utility,  able  to  fill  but  a  narrow  place  in  the 
life  of  their  community,  and  having  an  education  but  very  little  better  than 
that  acquired  by  the  majority  of  skilled  laborers." 


IX.     SUMMARY. 

Replies  to  the  questionarire  letter  were  received  from  instructors 
of  mathematics  and  of  professional  subjects,  and  from  practicing 
engineers.  A  majority  of  the  replies  favor  entrance  examinations 
in  mathematics  for  all  students  while  some  very  grave  objections 
are  raised  to  such  a  practice,  but  there  is  a  general  demand  for  better 
preparation,  especially  in  algebra.  A  short  review,  together  with 
tests,  has  been  found  a  good  substitute  in  several  institutions.  Funda- 
mental principles  and  the  elements  of  the  subjects  taught  are  con- 
sidered of  primary  importance,  and  in  these  are  also  found  the 


96  MATHEMATICS  FOR  ENGINEERS. 

greatest  deficiencies,  as  remedies  for  which  greater  intensity  of 
work  and  less  extension  in  subject  matter,  together  with  a  vitaliza- 
tion  of  the  work  are  advised.  The  object  of  analytic  geometry  is 
to  furnish  the  student  with  a  new  mathematical  language  rather 
than  information  concerning  any  particular  class  of  curves,  and 
the  introduction  of  the  elements  of  the  calculus  in  connection  with 
analytic  geometry  is  favored  in  a  majority  of  the  replies.  Inability 
to  analyze  problems  and  to  use  the  mathematical  language  learned 
are  the  chief  general  difficulties  reported,  and  frequent  application 
of  the  principles  learned  and  teaching  for  future  use  may  be  helpful 
as  remedies.  It  is  deemed  advisable  to  teach  engineering  students 
in  separate  classes,  making  some  use  of  problems  met  in  actual 
engineering  work  for  the  purpose  of  creating  interest,  but  special 
texts  are  not  strongly  favored.  There  is  a  demand  for  work  in 
computation,  with  especial  attention  to  degrees  of  accuracy.  Little 
attention  is  paid  to  questions  concerned  with  modes  of  instruction, 
showing  the  need  of  professionally  trained  teachers.  Engineering 
students  are  thought  to  be  more  thoro  and  painstaking  than  literary 
students  in  general. 


CHAPTER  V. 

PRESENT  NEEDS  AND  TENDENCIES. 

I.  CONTENTS  OF  CHAPTER. 

Various  questions  regarding  the  needs  of  the  work  in  mathe- 
matics for  freshmen  engineers  and  the  trend  of  thought  of  the 
present  day  concerning  it  have  arisen  in  different  connections  in  the 
preceding  chapters.  The  discussion  of  such  questions  has  been  held 
in  abeyance  until  the  present  chapter  when  each  can  be  taken  up  in 
its  entirety  in  the  light  of  the  accumulated  information.  Besides 
the  statements  found  in  the  previous  chapters,  use  will  be  made  of 
the  few  works  published  on  the  subject,  articles  in  the  various 
journals,  proceedings  of  meetings  of  societies,  personal  interviews 
and  personal  observations.  As  far  as  possible  the  various  sug- 
gestions for  improvement  have  been  tested  by  the  writer's  own 
experience. 

II.    PREPARATORY  MATHEMATICS. 

If,  as  has  been  said,  a  man's  education  should  begin  with  his 
great  grandparents,  then  surely  the  mathematical  education  of  the 
student  of  engineering  should  begin,  at  the  very  least  in  the  high 
school.  The  general  demand  for  better  mathematical  preparation  of 
the  students  entering  the  freshman  year  makes  the  question  of  pre- 
paration an  essential  part  of  our  discussion. 

i)  NATURE  OF  WORK  NEEDED. 

The  preparatory  mathematics  of  prospective  engineering  stu- 
dents should  be  taught  with  a  view  to  its  utilization.  Especially 
is  this  true  of  algebra,  where  each  principle  should  be  studied  for 
its  future  use,  all  the  work  being  grouped  around  the  equation  as  the 


98  MATHEMATICS  FOR  ENGINEERS. 

predominating  feature.  The  work  in  geometry  may  profitably  be 
made  more  observational,  and  a  greater  number  of  practical  prob- 
lems introduced,  especially  in  solid  geometry.1  Special  classes 
in  high  school  mathematics  for  prospective  students  of  engineering 
are  impracticable  but  all  the  work  should  be  given  with  a  view  to 
its  possible  application.  The  suggestion  that  the  collegiate  mathe- 
matics be  given  from  the  standpoint  of  utility  applies  even  more 
forcibly  to  preparatory  work.2 

(i)  Algebra  to  be  Treated  as  a  Study  of  Equations. 

^  The  writer  had  the  opportunity,  a  few  years  ago,  of  teaching 
beginning  high  school  algebra  simultaneously  to  two  sections  of 
about  equal  industry  and  ability.  With  one  the  text  was  followed 
quite  closely  and  the  results  showed  the  usual  average  of  interest 
and  achievement.  The  other  section,  as  an  experiment,  was  given 
an  extended  course  in  equations,  all  other  work  being  taken  up  as 
necessary  adjuncts.  The  pupils  were  introduced  to  the  equation 
through  problems  of  arithmetic  and  were  shown  (not  told)  that 
algebra  is  only  an  extension  of  arithmetic.  This  was  done  somewhat 
in  the  following  manner:  two  or  three  problems  were  given  on 
the  relation  between  the  sides  of  the  right  triangle  which  had  been 
learned  in  arithmetic,  for  example,  finding  the  height  of  a  tree  which 
had  been  blown  down  in  such  a  maner  that  it  is  still  in  contact  with 
the  stump,  which  is  six  feet  high,  and  the  end  reaches  the  ground 
eight  feet  from  the  foot  of  the  stump.  Then  a  problem  was  intro- 
duced regarding  a  tree  broken  off  in  a  similar  manner  but  where 
the  unknown  number  had  to  be  used  implicitly,  as  where  the  length 
of  the  tree  and  the  distance  of  its  top  from  the  base  of  the  stump 
were  given,  the  length  of  the  part  broken  off  being  asked  for. 
Where  the  results  were  whole  numbers  some  solutions  were  always 
obtained,  but  for  other  cases  none  could  be  found.  This  brought 
out  the  necessity  of  more  general  tools  for  the  solution  of  problems 
and  the  need  of  the  "unknown  quantity."  At  first  the  equation  was 
used  only  to  solve  for  the  unknown  quantities.  These  were  never 
represented  by  X,  Y,  or  Z  but  by  L  for  length,  W  for  weight,  etc. 
After  some  time  the  second  great  use  of  the  equation  was  taken 

1  Chap.  IV,  p.  81. 
»Chap.  IV,  p.  81. 


PRESENT  NEEDS  AND  TENDENCIES.  99 

up;  namely,  to  show  the  relation  between  various  quantities.  The 
graph  in  two  variables  was  now  introduced  to  give  a  geometric  pic- 
ture of  this  relation,  and  the  connection  between  the  graph  and 
the  equation  established.  A  number  of  simple  graphs  were  made 
either  for  quantities  which  could  be  read  directly  from  the  instru- 
ments presented  before  the  class,  as  from  levers,  the  apparatus  ex- 
hibiting Boyl's  Law,  etc.,  or  for  those  which  could  be  easily  computed, 
as  for  interest,  change  of  United  States  to  English  money,  etc.  A 
little  later  this  was  extended  so  as  to  include  any  available  data, 
taken  mostly  from  physics.  Two  graphs  drawn  with  the  same 
coordinate  axes  furnished  an  introduction  to  simultaneous  equa- 
tions, where,  of  course,  only  linear  equations  were  at  first  con- 
sidered. Only  one  or  two  illustrations  as  to  mode  of  introducing 
the  various  topics  ordinarily  taught  in  algebra  can  be  given.  Frac- 
tions were  taken  up  when  literal  quantities  were  met  with  in  the 
denominators  of  fractions,  radicals  when  such  equations  as  X2  =  8 


had  to  be  solved,  and  later  on  VX  —  4  —  3  yX  -}-  2=  ~\/X  —  i, 
etc.  The  modes  used  to  introduce  the  various  topics  may  have  been 
very  artificial  but  they  secured  the  desired  results  ;  they  created 
a  great  interest  in  the  work  and  produced  at  the  end  of  the  year  a 
class  capable  of  using  algebraic  equations  and  of  interpreting 
results  obtained  from  its  solution  in  a  very  satisfactory  manner. 

2)  DEFICIENCIES  FOUND. 

The  teaching  of  geometry  has  elicted  relatively  light  criticism.3 
How  far  this  is  due  to  good  teaching  and  how  far  to  the  fact 
that  the  future  work  is  not  largely  dependant  upon  geometry  can 
not  readily  be  ascertained.  But  in  algebra  deficiencies  abound  ;  the 
principal  ones  relating  to  surds,  exponents,  quadratics,  imaginaries 
and  the  manipulation  of  algebraic  symbols.  The  one  chief  objection, 
however,  to  the  high  school  algebra  as  now  taught  is  that  the  students 
can  not  apply  it* 

3)  CAUSE  OF  DEFICIENCIES. 

There  are  really  two  causes  of  deficiencies  ;  those  due  to  the 
curriculum  and  the  school  management,  and  those  due  to  the  teacher. 

8  Chap.  IV,  p.  71. 
•  Chap.  IV,  p.  91. 


woi 

sub 
of, 


ioo  MATHEMATICS  FOR  ENGINEERS. 

(*)  School  Management. 

The  school  management  is  responsible  for  four  great  causes  of 
poor  preparation: 

Fully  75%  of  the  difficulties  are  due  to  the  long  interval  between 
the  work  in  algebra  in  the  high  school  and  that  of  the  freshman 
year. 

Students  are  not  always  required  to  attend  classes  to  obtain 
the  credits  presented  for  entrance.  This  arises  in  two  ways ;  first 
when  students  take  all  of  the  work  in  algebra  at  an  unaccredited 
school,  and  then  go  to  one  which  is  accredited  without  having  to 
"prove"  their  grades  at  the  latter,  and  secondly,  where  a  student 
is  permitted  to  do  part  of  the  work  outside  of  the  regular  class 
and  "pass  an  examination"  on  the  work  so  as  "to  graduate  with  his 
class,"  or  for  some  similarly  trivial  reason. 

In  some  cities  work  in  mathematics,  especially  algebra,  is  al- 
loted  to  various  teachers,  who  have,  as  a  rule,  been  selected  for 
their  ability  in  other  subjects  rather  than  in  mathematics. 

The  last  hindrance  to  be  mentioned  is  that  if  the  high  school 
is  too  small  to  have  a  physical  director,  the  mathematics  teacher  is 
often  the  foot-ball  and  base-ball  coach.  One  never  hears  of  a 
teacher  of  Latin  selected  because  he  can  coach  the  athletic  teams. 

(u)  The  Teacher. 

Teachers  who  devote  all  their  time  to  teaching  mathematics 
may  still  have  poor  preparation  or  may  feel  little  enthusiasm  in 
their  work,  and  will  consequently  produce  poor  results.  Most  of 
the  work  is  taught  as  a  set  of  heterogeneous  topics  bearing  no  re- 
lation to  each  other,  and  with  the  expectation  that  each  will  be  "placed 
upon  the  shelf"  as  soon  as  "completed."  In  no  way  is  the  work  given 
with  a  view  to  future  use.  No  wonder  that  the  students  can  not 
apply  radicals  or  find  factors  when  solving  quadratic  equations.  On 
the  other  hand,  an  enthusiastic  mathematician  is  liable  to  make  the 
work  too  theoretical  and  to  over  emphasize  the  analytic  side  of  the 
subject.  In  either  case  the  pupils'  interest,  the  one  chief  element 
of  success  in  secondary  work,  is  not  secured. 

WMMtoi 

4)  SUGGESTED  REMEDIES. 

The  engineering  colleges  can  aid  in  the  crusade  for  better  teach- 
ing of  preparatory  mathematics  by  demanding  utilitarian  mathemat- 


PRESENT  NEEDS  AND  TENDENCIES.  IQI 

ics  from  the  high  schools  and  by  showing  its  educational  value.  To 
produce  results  these  demands  must  be  more  than  mere  suggestions 
or  requests. 

(i)  Requirement  of  Entrance  Examinations  in  Mathematics. 

One  way  of  making  this  demand  is  to  require  entrance  exami- 
nations in  mathematics  of  all  students.5  Besides  necessitating  better 
work  on  the  part  of  the  high  schools,  such  examinations  would  result 
in  reviews  of  the  high  school  subjects  before  taking  up  the  collegiate 
work  and  would  also  acquaint  the  instructors  with  the  students' 
weaknesses.6  To  the  objection  that  this  requirement  would  not  be 
fair  to  the  other  departments,  it  may  be  said  that  this  system  is  in 
force  with  the  Department  of  English  in  well  recognized  institutions. 
To  the  added  objection  that  a  great  deal  of  disturbance  among  the 
accredited  schools  would  result,  the  reply  may  be  that  this  disturb- 
ance might  have  a  salutary  effect.7 

(it)  Substitute  for  Entrance  Examinations. 

To  obtain  the  beneficial  results  of  entrance  examinations  with- 
out their  objectionable  features  some  institutions  have  hit  upon  the 
plan  of  devoting  the  first  two  or  three  weeks  to  a  review  in  algebra 
interspersed  with  tests.8  For  those  who  are  seen  to  be  notably 
deficient  a  course  in  the  high  school  is  recommended,  and  to  those 
only  slightly  so,  further  work  in  review  is  given.  This  plan  gives 
the  students  the  necessary  review  and  saves  them  from  the  loss  of 
time  and  the  discouragement  of  failure.  All  reports  showed  that 
this  plan  has  proved  a  decided  success  wherever  tried. 

(Hi)  Demands  upon  the  High  Schools. 

Engineering  colleges  that  admit  graduates  of  high  schools 
without  examinations  should  insist  that  the  instruction  in  mathe- 
matics in  these  schools  be  given  by  special  teachers  of  that  subject, 
and  in  no  case  by  teachers  selected  because  of  their  ability  as 
athletic  coaches  or  on  other  irrelevant  grounds.  All  work  for  which 

s  64  per  cent  of  the  answers  to  the  questionaire  letter  favored  entrance  examinations 
in  mathematics.     Chap.  IV,  p.  71. 
•Chap.  IV,  p.  71. 
T  Chap.  IV,  p.  71. 
•  Chap.  IV,  p.  71- 


102  .MATHEMATICS  FOR  ENGINEERS. 

credit  is  given  should  be  taken  in  class.  A  review  of  algebra  dur- 
ing the  fourth  year  should  also  be  one  of  the  requirements  for  ex- 
emption from  entrance  examinations. 


III.    COLLEGIATE  MATHEMATICS. 

With  respect  to  the  collegiate  mathematical  instruction  of  stu- 
dents of  engineering  we  consider  i)  what  should  be  taught,  2)  how 
it  should  be  taught. 

1)  WHAT  SHOULD  BE  TAUGHT. 

Future  use  should  be  the  basis  of  the  mathematical  curriculum. 
Practicing  engineers  and  instructors  of  engineering  subjects  are 
best  qualified  to  speak  of  the  future  mathematical  needs  of  the  en- 
gineering student.  Consequently,  they  may  justly  claim  the  privilege 
of  designating  the  contents  of  the  mathematical  curriculum.  Such 
suggestions  from  the  professional  men  should  allow  for  time  to 
give  all  prerequisite  topics,  and  should  be  really  as  well  as  nomi- 
nally mathematical,  calling  for  not  the  mere  teaching  of  "rules  of 
thumb"  but  for  the  use  and  application  of  mathematical  principles. 

2)  How  IT  SHOULD  BE  TAUGHT. 

When  the  matter  to  be  taught  has  been  determined  it  is  the 
province  of  the  mathematical  department  to  consider  the  designated 
material  and  all  prerequisite  topics  without  extraneous  matter,  and 
to  organize  it  into  a  teachable  course.  The  duty  of  teaching  this 
course  to  the  very  best  advantage  then  also  falls  upon  the  mathe- 
matical instructors. 

3)  SPECIFIC  AND  GENERAL  NEEDS  AND  DIFFICULTIES. 

At  the  completion  of  the  first  year's  work  in  mathematics  the 
students  are  expected  to  show  mathematical  ability  of  two  distinct 
kinds,  specific  and  general ;  while  the  difficulties  encountered  also 
fall  under  these  two  heads.  An  example  of  the  first  would  be  the 
handling  of  radicals ;  one  of  the  second,  the  ability  to  think  mathe- 
matically. This  is  a  helpful  distinction,  especially  in  bringing  out 


PRESENT  NEEDS  AND  TENDENCIES. 


103 


the  opinions  of  educators  and  professional  men  interested  in  these 
questions,  and  accordingly  was  adhered  to  in  making  out  the  ques- 
tionaire  letter  of  Chapter  IV.9 

The  specific  requirements  and  difficulties  are  somewhat  the 
easier  to  discuss  and  will  be  taken  up  first.  It  will  also  be  found 
that  the  general  ones  can  often  be  reached  thru  the  specific  and 
disposed  of  while  taking  care  of  the  latter. 

4)  SPECIFIC  NEEDS  AND  DIFFICULTIES. 

The  specific  needs  and  difficulties  will  be  considered  under  the 
subjects  of  trigonometry,  algebra,  analytic  geometry  and  the  calculus 
both  to  expedite  the  discussion  and  also  because  most  of  the  institu- 
tions observe  these  distinctions.  Pedagogic  questions  relative  to  the 
improvement  of  the  work  will  be  considered  in  appropriate  con- 
nections. 

(t)   Trigonometry. — a)  Reason  for  Place  in  Curriculum. 

Since  only  18%  of  the  institutions  require  trigonometry  as  a 
preparatory  subject  it  can  not  be  omitted  in  considering  the  work 
of  the  freshman  year.10  Further,  the  subject  is  of  the  utmost  im- 
portance in  engineering  work  and  while  comparatively  easy,  it  pre- 
sents an  astonishing  number  of  failures,  and  that  in  the  elements.11 
For  these  reasons  it  requires  the  very  best  of  instruction  and,  as  a 
consequence,  some  institutions  which  admit  without  examination  in 
other  mathematical  branches  require  one  in  trigonomery.12 

b)  Needs. 

A  thoro  grasp  of  the  meaning  of  the  trigonometric  functions 

and  their  use  in  the  solution  of  the  right  triangle  is  really  the  one 

thing  needed.13      Besides  this  the  student  needs  to  know  some  few 

relations  existing  between  the  functions,  to  be  able  to  solve  a  right 

triangle  placed  in  any  position,  to  pick  out  the  triangles  in  a  con- 

•  Chap.  IV,  pp.  67-70. 

"Chap.  II,  p.  43. 

u  An  instructor  in  railroad  engineering  said  he  had  "some  young  men  who  have 
been  considered  as  very  bright  in  the  calculus  but  who  can  make  no  headway  because 
they  have  not  the  foundations  of  trigonometry.  They  have  no  grasp  of  the  trigonometric 
functions  and  their  relations." 

"Chap.   Ill,  p.   54- 

13  Chap.  IV,  p.  74- 


104 


MATHEMATICS  FOR  ENGINEERS. 


figuration  essential  to  his  problem,  and  to  solve  oblique  triangles  by 
use  of  formulas.  Advanced  topics  as  hyperbolic  functions,  De 
Moivre's  Theorem,  etc.,  are  of  little  value  to  the  engineering  stu- 
dent.14 

c)  Deficiencies. 

The  most  marked  deficiencies  are  found  in  the  topics  just  men- 
tioned as  of  most  importance.  This  is  probably  due  to  the  fact 
that  the  elements  are  called  into  play  the  most  frequently  and  points 
out  the  need  of  special  care  in  teaching  the  fundamental  definitions 
and  properties  of  the  trigonometric  functions.  The  use  of  identities, 
especially  those  relating  to  functions  of  2  X  and  l/2  X,  inverse  func- 
tions, circular  functions  and  application  of  the  definitions  of  the 
trigonometric  functions  are  also  found  to  give  serious  difficulty 
while  still  another  grave  deficiency  is  the  inability  to  see  what 
triangles  are  to  be  solved  and  to  solve  them  in  positions  other  than 
the  standard  one  with  one  side  horizontal." 

d)  Suggestions  for  Improvement. 

In  consequence  of  the  deficiencies  found,  the  work  is  to  be 
simplified  and  confined  mainly  to  the  elements.  The  functions  should 
be  defined  from  the  ratio  standpoint  and  the  definitions  constantly 
reviewed.  To  a  great  number  of  students  who  have  learned  the 
functions  with  reference  to  the  unit  circle  the  sine  means  the  vertical 
line  opposite  an  acute  angle  in  a  triangle,  etc.  The  meaning  of  the 
trigonometric  functions  can  be  made  clear  by  the  solution  of  numer- 
ous right  triangles,  which  should  be  placed  in  various  positions  and 
not  in  any  standard  one.  Constant  application  of  theoretical  re- 
sults to  a  variety  of  real  problems,  of  which  a  rich  store  lies  at  hand, 
is  also  indispensable. 

e)  Trigonometry  to  be  Taught  from  the  Utilitarian  Standpoint. 

The  utility  of  no  study  is  more  obvious  to  the  engineer  than 
that  of  trigonometry.  Use  can  be  made  of  this  fact  to  arouse  in- 
terest and  action  on  the  part  of  the  student.  The  very  first  lesson 
is  the  best  place  to  bring  out  this  utilitarian  idea  in  order  to  make 
a  lasting  impression  and  this  lesson  can  be  made  to  contain  a  large 

M  Chap.  IV,  p.   73. 
u  Chap.  IV,  p.  74. 


PRESENT  NEEDS  AND  TENDENCIES. 


105 


part  of  the  elements  of  the  subject.  This  should  not  be  done  by  the 
formal  statement  that  trigonometry  will  be  a  most  necessary  part 
of  the  future  work  in  mathematics  and  physics  but  by  well  selected 
illustrative  problems,  such  as  finding  the  height  of  buildings,  flag 
poles  and  trees  by  using  lines  of  known  length  drawn  from  their 
bases  along  the  ground,  width  of  rivers  using  base  lines  along  their 
banks,  etc.  If  a  surveyor's  transit  is  not  available  for  these  first 
lessons  a  model  should  be  improvised.  As  another  means  to  keep 
up  interest  a  formal  problem  where  solution  will  be  within  reach  of 
the  students  at  the  end  of  two  or  three  weeks'  work  can  be  proposed 
on  one  of  the  first  days  to  be  kept  constantly  in  view  for  solution. 
The  finding  of  the  resultant  of  several  forces  passing  thru  a  point 
by  first  resolving  them  along  two  directions  at  right  angles  may 
be  cited  as  an  illustration.  For  the  solution  of  problems  on  the 
right  triangle  similar  to  those  mentioned  above  only  the  sine  and 
tangent  functions  are  necessary.  The  functions  of  any  angle  having 
been  defined  the  students  may  be  required  to  determine  the  values 
approximately  from  measurements,  or  they  may  be  provided  with 
tables  of  natural  functions  at  once.  The  instruction  should  be  con- 
tinued in  the  same  spirit  with  the  solution  of  the  oblique  triangle  as 
the  ultimate  goal. 

f)   Suggested  Course. 

In  accordance  with  the  above  it  is  advisable  to  devote  two 
hours  per  week  during  the  first  half  of  the  year  to  trigonometry. 
The  work  is  to  be  introduced  by  the  solution  of  a  great  number  of 
practical  problems  solvable  by  means  of  the  trigonometric  func- 
tions applied  to  the  right  triangle.  This  is  to  be  followed  by 
exercises  on  the  relation  between  the  functions,  and  also  the  func- 
tions of  two  or  more  angles,  leading  up  to  the  solution  of  the 
oblique  triangle.  Logarithms  are  taken  up  just  before  the  oblique 
triangle.  A  short  discussion  of  circular  functions  should  also  be 
included. 

(tt)  Algebra. — a)   Status  in  Engineering  Colleges. 

Of  all  the  mathematical  subjects  in  the  engineering  curriculum, 
algebra  is  in  the  most  chaotic  condition.  The  entrance  requirements 
vary  from  nothing  at  all  to  three  full  years,  with  a  corresponding 
variation  in  the  collegiate  work.  The  difference  which  is  most  to 


I06  MATHEMATICS  FOR  ENGINEERS. 

the  point  is  found  in  the  topics  given  under  the  head  of  College 
Algebra.  No  schematic  exhibition  of  the  subject  matter  can  be 
made  as  many  of  the  catalogs  do  not  state  what  topics  are  included 
under  College  Algebra  and  those  which  do,  present  too  great  a 
variety  for  comparison.  The  courses  seem  to  have  in  common  chiefly 
the  two  topics  of  the  Binomial  Theorem  and  the  theory  of  equa- 
tions, (the  latter  a  very  indefinite  term,  since  its  scope  is  rarely 
stated  fully),  and  the  failure  to  mention  any  review  of  previous 
work  even  in  quadratics.  As  to  the  rest  there  is  a  wide  range  of 
variation  for  which  no  definite  cause  can  be  assigned.  While  some 
topics  are  more  especially  useful  to  students  taking  a  particular 
course  (such  as  imaginaries  for  the  electrical  engineer),  this  special- 
ization is  not  recognized  by  institutions  offering  several  professional 
courses.  Neither  can  the  reason  be  that  algebra  is  given  as  a  cul- 
ture study.  The  many  serious  difficulties  encountered  in  algebraic 
reductions  by  students  in  advanced  mathematical  or  technical  work 
make  it  impossible  to  regard  algebra  otherwise  than  as  a  most  vital 
foundation  for  subsequent  work. 

b)  Needs  and  Difficulties. 

The  greatest  difficulty  is  experienced  in  the  handling  of  the 
fundamental  algebraic  operations,  with  which,  first  of  all,  the  stu- 
dent must  be  familiar.  Determinants,  permutations  and  combina- 
tions, and  series  are  of  minor  importance.  On  the  other  hand, 
graphical  solutions  of  equations,  graphical  representations  of  cer- 
tain series,  the  treatment  of  imaginaries  and  the  solution  of  equa- 
tions of  the  third  degree  call  for  special  attention.16 

c)  vSuggested  Improvements. — /)   Teach  Elements. 

The  first  aim  is  to  give  to  the  student  a  working  knowledge  of 
fundamental  principles.  Every  effort  should  be  made  to  improve  the 
preparation  given  by  the  high  school17  but  where  students  are  still 
deficient  remedies  must  be  applied  at  the  earliest  possible  moment 
so  that  they  may  not  feel  a  handicap  later.  The  review  of  the  main 
principles  of  the  high  school  algebra  as  previously  outlined  will  be 
found  helpful.18 

MChap.  IV,  p.  76. 
11  Chap.  V,  p.  101. 
u  Chap.  V,  p.  102. 


PRESENT  NEEDS  AND  TENDENCIES.  107 

77)  Principles  of  Algebra  to  be  Taken  Up  When  Needed. 

It  is  a  mistake  to  conduct  the  instruction  on  the  assumption 
that  students  know  and  remember  every  principle  previously  studied. 
Perhaps  the  professional  men  ask  too  much  of  the  instructors  of 
freshmen  mathematics  by  taking  too  much  for  granted,  while  the 
latter  in  turn  require  too  much  of  the  high  school  teachers.     Time 
can  often  be  saved  by  giving  preliminary  reviews  of  principles  in- 
volved in  new  work  to  be  taken  up.    The  plan  has  been  carried  so 
far  as  to  give  no  formal  course  in  college  algebra  but  to  take  up 
each  algebraic  topic  as  it  arises  in  connection  with  other  branches 
of  mathematics.    This  somewhat  novel  idea  offers  some  promise  of 
help  in  alleviating  difficulties  which  confront  the  teacher  of  fresh- 
man mathematics.20   The  objections  to  such  a  method  are  first,  that 
it  may  have  a  tendency  to  foster  "a  rule  of  thumb"  way  of  treating 
each  mathematical  process  as  applicable  to  but  a  single  case;  and 
second  that  the  student  does  not  learn  the  mathematical  principles 
but  only  a  way  of  connecting  some  mechanical  manipulation  with 
the  particular  process  in  question.     To  avoid  so  great  an  evil  the 
method  must  not  be  carried  to  extremes,  and  the  principles  should 
be  applied  in  several  different  phases  of  use.     In  small  sections 
taking  very  specialized  courses  this  plan  commends  itself  most  fav- 
orably.    Other  plans  of  a  more  radical  nature  are  those  in  which 
college  algebra  is  taken  in  connection  with  either  trigonometry  or 
analytic  geometry  in  such  a  way  as  to  bring  out  its  points  of  con- 
tact with  these  subjects,  and  its  graphic  aspect.21    As  the  portions 
of  college  algebra  that  are  really  advanced  are  little  used  before  tak- 
ing up  mechanics,  real  college  algebra  can  well  be  deferred  until 
after  analytic  geometry  and  the  calculus  have  been  taken,  the  alge- 
braic work  given  earlier  being  essentially  a  review  and  an  extension 
of  the  elementary  work. 

(wi)  Analytic  Geometry. — a)  As  a  Mathematical  Language. 

Since  engineering  students  must  use  algebra,  and  as  long  as 
there  exists  such  extreme  difficulty  in  applying  it,  collegiate  work 
in  algebra  should  not  be  given  them  as  a  culture  study.  Let  the 

»  Chap.  Ill,  p.  57. 

a  Frederick  L.  Emory,  Advanced  Algebra  in  Engineering  and  other  Colleges,  Proc. 
Soc..  for  Prom.  Eng.  Ed.,  vol.  Ill,  p.  104. 


io8  MATHEMATICS  FOR  ENGINEERS. 

instructor  teach  that  which  will  be  met  with  in  the  future  work  ;  for 
example,  in  radicals,  let  him  drill  on  a  systematic  use  of  the  three 
fundamental  rules  to  which  all  can  be  reduced  : 

a)  the  removal  of  radical  denominators, 

b)  simplification  of  expressions  where  the  number  under  the 
radical  is  a  perfect  power  indicated  by  a  factor  of  the  index,  as 


c)  simplification  of  expressions  containing  factors  of  which 
the  radical  root  can  be  extracted,  as  V12- 

If  the  students  aftenvards  offer  a  result  such  as  V8  in  the 
solution  of  a  problem  he  deserves  a  reduction  in  grade. 

The  following  few  rules  are  quite  effective  in  a  course  in  col- 
lege algebra: 

a)  give  a  short  review  of  exponents,  radicals,  quadratics,  etc., 

b)  introduce  new  topics  by  oral  quizzes  on  the  earlier  princi- 
ples that  will  be  involved, 

c)  insist  that  principles  previously  learned  be  employed  when- 
ever possible, 

d)  demand  an  analysis  of  each  problem  and  an  understanding 
of  each  formula  used,  without,  however,  requiring  the  development 
off-hand. 

Chapter  III  contains  several  suggestions  for  making  both  the 
position  and  the  object  of  college  algebra  definite  in  the  curriculum. 

*«)  Analytic  Geometry.  —  a)  As  a  Mathematical  Language. 

When  the  student  takes  up  the  study  of  analytic  geometry, 
whether  designated  by  that  formal  title  or  not,  he  meets  with  a  new 
language  just  as  much  as  when  he  begins  the  study  of  French  or 
German.  There  must,  of  course,  be  some  medium  to  which  to  apply 
this  new  language  while  learning  it.  Until  very  recently  the  conic 
sections  have  been  universally  selected  as  this  medium  and  often  the 
learning  of  the  properties  of  the  conies  has  been  the  main  business 
of  the  course,  thereby  turning  it  into  a  mere  memorization  of  a  col- 
lection of  isolated  properties  and  formulas  of  the  curves  and  losing 
track  of  the  analytic  mode  of  attack,  which  should  be  its  chief  aim.22 

22  "Too  much  space  should  not  be  given  to  the  investigation  of  the  properties  of  the 
conic  sections,  --  etc."  Mansfield  Merriman,  Teachers  and  Text-books  in  Mathematics  in 
Technical  Schools,  Proc.  Soc.  for  the  Prom,  of  Eng.  Ed.  vol.  II,  p.  95. 


PRESENT  NEEDS  AND  TENDENCIES, 


109 


It  is  wholly  immaterial  what  medium  is  used  so  long  as  it  supplies 
the  requirement  of  teaching  the  language,  and  interests  the  students 
in  its  use.  If  in  addition  to  this  it  can  acquaint  the  student  with  some 
curves  met  with  in  his  future  work,  that  is  an  additional  advantage. 
Any  problems  in  mechanics  or  geometry  possessing  these  qualifica- 
tions are  suitable,  and  conies  are  to  be  treated  only  as  varieties  of 
curves  in  general. 

b)   Illustrative  Problems. 

The  following  are  a  few  illustrations  of  problems  of  the  nature 
just  mentioned  which  have  been  found  useful  by  the  writer: 

1)  Find  the  equation  of  a  projectile  given  an  initial  horizontal 
motion. 

2)  Ditto  if  thrown  at  an  angle  to  the  horizontal. 

3)  Prove  that  parallel  rays  can  be  brought  to  a  focus  by  a  para- 
bolic reflector. 

4)  Proof  of  the  method  of  inscribing  a  parabola  in  a  rectangle 
as  used  in  mechanical  drawing  constructions. 

5)  Ditto  for  hyperbola. 

The  following  are  examples  of  curves  for  study: 


i  — e 


no  MATHEMATICS  FOR  ENGINEERS. 

c)  Other  Devices. 

It  is  in  the  application  and  in  the  appreciation  of  the  analytic 
language  that  the  most  frequent  difficulties  arise.23  Besides  the  prob- 
lems mentioned  above  several  devices  have  been  found  useful,  one 
of  which  is  the  analytic  proof  of  theorems  found  in  Euclilean  geom- 
etry. Such  problems  as  the  following  may  be  assigned  as  a  part 
of  the  following  day's  work : 

1)  "From  the  middle  point  D  of  the  base  of  a  right  angled 
triangle  ABC,  DE  is  drawn  perpendicular  to  the  hypotenuse  BC. 
Prove  that  BE2  —  EC2  =  ABV 

2)  "CO  is  the  line  drawn  from  the  center  of  a  circle  to  any 
point  of  the  chord  AB.    Prove  OC2  =  OA2  —  AC  X  CB."2* 

For  off-hand  solutions  in  the  class  room  the  most  efficient  are 
the  simpler  ones  involving  the  principles  at  hand,  such  as,  to  prove 
that  the  perpendiculars  from  the  vertices  of  a  triangle  upon  the  op- 
posite sides  meet  in  a  point. 

For  bringing  out  the  true  analytic  spirit  the  "hammer  and  tongs 
process"  may  be  used  in  which  the  unknown  relations  are  given 
analytic  expressions  however  long  these  may  be,  and  then  applied 
as  tests  to  the  theorem  to  be  proved.  In  illustration  the  problems 
above  stated  may  be  cited.  The  expressions  for  the  quantities  in  the 
formulas  are  worked  out  and  with  these  the  truth  of  the  statement 
verified  or  disproved.  Such  work  is  often  very  long  but  is  of  great 
assistance  in  bringing  the  student  to  see  the  power  of  the  analytic 
expressions  of  geometric  facts. 

A  full  understanding  of  the  idea  of  parameter  is  a  decided  aid 
to  a  clear  grasp  of  problems  in  which  loci  are  found  by  eliminating 
a  parameter  from  two  or  more  equations.  The  parameter  should 
be  taken  up  early  and  its  meaning  for  any  particular  equation  dis- 
cussed some  time  before  the  loci  problems  depending  upon  its  elim- 
ination are  attempted,  as  otherwise  the  idea  of  the  parameter  is  lost 
in  following  the  mechanical  process  of  elimination.  The  parameter 
is  also  useful  in  illustrating  families  of  curves  and  in  showing  their 
variations.  In  aiding  a  class  to  see  the  deductions  of  the  equation 
of  a  locus  when  the  parameter  was  a  point  (X',  Y')  the  writer  hit 

»  Chap.  IV,  p.  77- 

84  Sanders,   Elements  of  Plane  Geometry.     New  York,   1901.  p.   216. 


PRESENT  NEEDS  AND  TENDENCIES.  m 

upon  the  mechanical  device  of  using  red  chalk  for  marking  the  locus 
in  the  graph  and  also  for  the  corresponding  X'  and  Y'  coordinates 
in  the  equation  of  the  required  locus.  At  least  90%  saw  at  a  glance 
the  relation  between  the  graph  of  the  locus  and  its  equation.  In 
fact,  before  the  solution  was  fully  written  out,  the  young  man  who 
had  asked  for  it  exclaimed,  "I  could  have  done  it  if  I  had  had 
colored  pencils."  The  above  incident  gives  an  excellent  illustration 
of  the  value  of  the  maxim  "Teach  through  the  eye."25 

d)  Needs  and  Difficulties. 

The  principal  difficulties  naturally  arise  in  the  same  topics  as 
do  the  greatest  needs.  Besides  the  one  great  need  of  learning  the 
analytic  language  a  very  few  special  ones  may  be  found,  of  which 
loci  problems,  intersection  of  curves,  tangency  and  the  sketching 
of  curves  from  the  equation  ought  to  be  mentioned.26  Regular  prac- 
tice in  curve  sketching  and  the  use  of  equations  in  their  simplest 
form  with  slightly  varied  terms  like  the  following  has  been  found 
helpful  to  remedy  defects  in  the  latter  case: 


I) 

y2=  I2x 

2) 

x2  =  I2y 

3) 

y2  =  —  I2x 

4) 

x2  =  —  I2y 

5) 

y2=  12  (x  —  3) 

6) 

x2=i2(y-U3) 

(iv)  The  Calculus. — a)   Principles  Taught. 

Of  late  years  the  elements  of  the  calculus  have  been  introduced 
more  or  less  into  the  freshman  work  by  various  institutions.27  The 
general  idea  is  to  acquaint  the  student  with  the  methods  of  the  cal- 
culus rather  than  to  enter  into  the  theory  to  any  extent;  the  formal 
course  being  delayed  until  the  second  year.  Principles  are  developed 
by  problems  and  graphs,  and  the  methods  of  the  calculus  are  used 
before  being  rigorously  proved.28  The  work  is,  as  a  rule,  limited 

M  J.  W.  A.  Young,  The  Teaching  of  Mathematics  in  the  Elementary  and  the  Second- 
ary Schools.     New  York,   1907,  p.  no. 
*"  Chap.  IV,  p.  78. 
»  Chap.  Ill,  p.  65. 
"Chap.   Ill,  p.   S3- 


H2  MA  THEM  A  TICS  FOR  ENGINEERS. 

to  problems  on  tangency  and  on  maxima  and  minima,  the  former  of 
which  affords  a  natural  means  for  its  introduction. 

b)  Advantages. 

This  early  introduction  to  the  use  and  methods  of  the  calculus 
and  an  acquaintance  with  its  powerful  machinery,  is  expected  to 
meet  the  demand  for  the  calculus  in  practical  problems.  It  gives 
the  most  powerful  and  the  most  suitable  instrument  for  solving  a 
great  number  of  problems  that  arise  in  algebra,  analytic  geometry 
and  physics,  in  the  first  year  or  early  in  the  second  year.  Further, 
the  calculus  will  not  only  be  thus  applied  more  extensively  but  will 
also  be  studied  during  a  materially  longer  period  of  time,  which  is 
a  decided  advantage.  The  interest  of  the  student  is  also  materially 
enhanced  when  the  work  is  directed  along  the  lines  of  what  can  be 
done  with  the  calculus  rather  than  with  the  development  of  formulas 
and  their  rigorous  proofs.  Such  an  introduction  of  the  calculus 
should  make  the  student  feel  that  it  is  not  a  wholly  new  and  distinct 
branch  of  mathematics  but  only  a  continuation  and  natural  out- 
growth of  what  has  preceded.  While  the  data  previously  recorded 
do  not  show  any  marked  tendency  actually  to  make  this  early  intro- 
duction to  the  calculus,  yet  the  trend  of  opinion  is  really  in  its  favor 
since  all  who  reported  having  tried  the  plan  had  found  it  a  success.29 

(v)  Combined  Courses  in  Algebra,  Coordinate  Geometry  and  Ele- 
ments of  the  Calculus. 

Coordinate  geometry  may  be  made  the  basis  of  a  combined 
course  in  algebra,  coordinate  geometry  and  the  elements  of  calculus 
to  occupy  two  or  three  hours  the  first  half  and  four  or  five  hours 
the  second  half  year.  After  a  two  or  three  weeks'  review  of  the 
most  needed  preparatory  algebra,  the  graphic  representation  of  the 
algebraic  function  is  taken  up,  where  such  problems  as  finding  points 
of  intersection  are  solved  and  such  elementary  notions  as  determin- 
ing the  distance  between  two  points  are  developed.  Algebraic  prin- 
ciples are  thereafter  taken  up  as  occasion  demands.  A  full  treat- 
ment of  the  right  line  and  circle  follows.  A  study  of  any  suitable 
interesting  curves,  including  conies,  completes  the  work  in  plane 
analytic  geometry,  where  the  chief  aim  is  to  familiarize  the  student 

*»  Chap.  IV,  p.  80. 


PRESENT  NEEDS  AND  TENDENCIES.  n3 

with  this  new  mathematical  language  and  to  train  him  in  its  use. 
Special  stress  is  placed  upon  loci  problems  throughout.  The  meth- 
ods of  the  calculus  are  to  be  used  in  the  tangent  problems,  the  find- 
ing of  maxima  and  minima  of  functions,  and  all  other  problems 
which  are  best  treated  by  such  methods.  No  introduction  to  the 
calculus  other  than  the  application  of  its  elementary  principles  to 
problems  is  to  be  attempted.  Differentiation  and  integration  are  to 
be  carried  on  simultaneously.  If  time  permits  geometry  of  three 
dimensions  may  complete  the  course,  but  this  can  be  delayed  with 
advantage  until  the  study  of  partial  derivatives. 

(vi)   Work  in  Computation. — a)   Mode  of  Presentation. 

The  need  of  drill  in  computation  is  evident,30  the  only  question 
being  as  to  the  mode  of  presentation.  Two  modes  suggest  them- 
selves :  one,  the  formulating  of  such  work  into  a  special  course  as 
has  been  done  by  some  institutions,31  and  the  other  its  distribution 
throughout  the  mathematical  course.  While  the  former  plan  would 
emphasize  the  necessity  of  such  drill  the  latter  is  rather  to  be  pre- 
ferred as  it  will  show  the  necessity  of  accuracy  at  all  times  and  en- 
hance the  value  of  the  drill  by  extending  it  over  a  much  longer 
period.  Besides  the  assignment  of  special  exercises  in  computation 
as  occasion  arises  it  has  been  found  to  be  quite  helpful  to  require  all 
final  expressions  to  be  reduced  to  their  simplest  form  for  computa- 
tion ;  that  is  so  as  to  involve  the  minimum  amount  of  arithmetical 
operations.  For  example,  let  radicals  always  be  reduced  to  the  form 

-r-\j  c   which  is  the  simplest  for  computation  by  the  use  of  tables. 

Let  the  student  employ  tables,  even  though  unacquainted  with  the 
theory  involved  in  their  construction. 

By  far  the  most  potent  aid  in  achieving  accuracy  is  the  applica- 
tion of  checks  upon  the  work,  which  is  always  possible  except  in  rare 
instances.  To  impress  the  need  of  such  checks  make  them  a  part  of 
the  problems  assigned  and  to  emphasize  it  still  further  change  the 
data  of  the  problems  in  the  text  or  assign  outside  ones  in  which 
accuracy  is  the  main  requirement.  The  importance  of  this  phase  of 
the  problem  work  for  students  of  engineering  can  not  be  overesti- 

»°  Chap.  IV,  p.  85. 

31  Chap.  Ill,  pp.  62-63. 


MATHEMATICS  FOR  ENGINEERS. 

mated.  An  added  advantage  of  no  mean  importance  gained  from 
such  a  system  of  checks  is  the  drill  in  insight  and  ingenuity  which 
it  gives. 

b)  Degrees  of  Accuracy. 

Of  nearly  equal  importance  with  accuracy  itself  is  the  ability 
to  understand  the  degree  of  accuracy  of  which  a  problem  is  capable. 
It  is  worse  than  useless  to  permit  a  student  to  carry  out  computa- 
tions to  decimal  places  beyond  those  for  which  the  data  are  known 
to  be  correct.  The  three  following  problems,  selected  from  differ- 
ent books  on  trigonometry,  may  serve  as  illustrations  showing  how 
a  much  greater  degree  of  precision  is  demanded  in  the  result  than 
is  found  in  the  data  if  the  latter  are  to  be  considered  as  having  the 
degree  of  precision  ordinarily  met  with : 

1 )  The  height  of  a  house  subtends  a  right  angle  at  a  win- 
dow on  the  other  side  of  the  street ;  and  the  elevation  of  the 
top  of  the  house  from  the  same  place  is  60°.    The  street  is  30  ft. 
wide.    How  high  is  the  house? 

The  result  is  given  as  69.282  ft.  This  result,  professing  to  ex- 
press the  height  of  the  house  within  a  thousandth  of  a  foot,  implies 
that  the  width  of  the  street  has  also  been  measured  with  this  degree 
of  accuracy.  This  is  not  customary  and  not  even  readily  possible. 
To  do  so  it  would  be  necessary  that  the  sides  of  the  street  them- 
selves be  fixed  within  a  possible  error  decidedly  under  a  thousandth 
of  a  foot,  and  that  measuring  instruments  of  corresponding  delicacy 
be  employed  in  measuring  the  width  of  the  street.  In  practice  these 
conditions  are  never  fulfilled.  A  good  question  would  have  been 
to  ask  for  the  height  in  feet  to  within  an  error  of  I  ft.  or  of  I  in. 

2)  From  the  top  of  a  hill  I  observe  two  mile-stones  on 
the  level  ground  in  a  straight  line  before  me  and  I  find  their 
angles  of  depression  to  be  respectively  5°  and  15°  ;  find  the 
height  of  the  hill. 

The  result,  given  as  228.6307  yds.,  professes  to  give  the  height 
of  the  hill  within  one  ten-thousandth  of  a  yard,  or  within  about 
1/250  in.,  which  is  much  less  than  the  presumable  error  in  the  plac- 
ing of  the  mile-stones,  and  which  also  implies  that  the  height  of  the 


PRESENT  NEEDS  AND  TENDENCIES.  115 

hill  is  a  quantity  that  can  be  determined  to  within  the  diameter  of 
a  grain  of  sand. 

3)  A  ladder  40  ft.  long  reaches  a  window  33  ft.  high  on 
one  side  of  the  street.  Its  foot  being  at  the  same  point,  it  will 
reach  a  window  21  ft.  high  on  the  opposite  side  of  the  street. 
Find  the  width  of  the  street. 

The  result  is  given  as  56.649  ft.,  to  which  the  same  criticism  is 
applicable  as  to  problem  i). 

When  accuracy  to  two  decimal  places  is  meant  such  expressions 
as  46  in.  ought  never  to  be  allowed  but  46.00  in.  insisted  upon.  Both 
from  the  engineering  standpoint  and  from  that  of  general  mathe- 
matical accuracy  these  conditions  should  be  considered  in  all  compu- 
tations. 

5)  GENERAL  NEEDS  AND  DIFFICULTIES. 

Several  needs  and  difficulties  of  a  general  nature  have  already 
been  considered  in  connection  with  the  various  branches  of  study. 
The  following  are  further  general  phases  which  have  not  been 
touched  upon: 

(*')  Cooperation  with  the  Professional  Departments. 

The  success  of  the  instructor  of  mathematics  can  be  greatly 
enhanced  if  the  instructors  of  the  professional  branches  will,  from 
time  to  time,  supply  him  with  lists  of  mathematical  difficulties  en- 
countered by  their  students  and  with  the  suggestions  as  to  the  prep- 
aration needed  for  their  respective  courses.  It  is  not  meant  that 
the  department  of  mathematics  should  duplicate  the  work  of  the 
professional  departments  in  any  sense,  but  only  supplement  it  so  as 
to  increase  the  efficiency  of  the  work  of  both  departments.  It  is 
not  meant  to  teach  a  special  "rule  of  thumb"  for  each  particular 
thing  to  be  taken  up  in  the  future  but  that  the  principles  and  results 
that  will  be  necessary  later  are  to  be  emphasized,  in  precisely  the 
same  way  as  the  instructor  of  trigonometry  emphasizes  certain  prin- 
ciples and  results  to  be  used  later  in  the  calculus,  mechanics,  etc.  In 
this  day  of  correlation  it  seems  surprising  that  definite  steps  have 
not  been  taken  to  any  great  extent  in  the  direction  just  mentioned. 
The  special  courses  in  mathematics  outlined  in  Chapter  III  have 


1 1 6  M  A  THEM  A  TICS  FOR  ENGINEERS. 

been  arranged  more  or  less  according  to  this  plan,  it  is  true,  and 
one  is  based  altogether  upon  the  work  to  be  taken  in  the  professional 
departments  during  the  last  two  years  but  the  plan  has  received  no 
wide  recognition.  Surely  such  cooperation  of  the  departments  would 
prove  greatly  beneficial  by  increasing  the  interest  of  the  students  in 
their  mathematical  work  thus  having  its  service  and  utility  pointed 
out,  by  remedying  much  of  the  weakness  of  the  students  in  applying 
their  mathematics  to  the  work  of  the  professional  courses^  and  finally 
by  fostering  more  congenial  relations  between  the  mathematical  and 
professional  departments. 

(it)  Problems— z)  Their  Use. 

Every  principle  studied  must  be  illustrated  by  a  sufficient  num- 
ber of  problems,  for  the  purpose  of  fixing  it  in  the  minds  of  the 
students  and  of  clearing  up  any  difficulties  which  may  have  arisen 
as  to  its  application.  To  emphasize  and  fix  the  mathematical  lan- 
guage many  short  problems  should  be  given  for  immediate  solution 
during  the  class  period :  the  shorter  the  problems  and  the  greater  the 
number,  the  better. 

b)  Kinds  of  Problems. 

Problems  are  like  puzzles:  if  too  simple  they  do  not  hold  the 
attention  and  if  too  difficult  they  extinguish  all  ambition  to  solve 
them.  Long  and  complicated  problems,  such  as  those  of  the  old 
"fox  and  hound"  found  in  elementary  algebras  only  a  few  years  ago, 
ought  to  be  discouraged.  Simplified  explanations  with  a  large  num- 
ber of  simple  problems  will  greatly  aid  in  preventing  students  from 
doing  things  by  rote  and  from  imitating  imperfectly  some  sample 
problem  which  has  been  worked  out  in  the  text-book.  In  order  to 
interest  the  students  the  problems  should  not  be  fantastic  but  related 
to  their  every  day  life  as  far  as  possible.  For  the  purpose  of  creat- 
ing further  interest  the  use  of  problems  connected  with  engineering 
work  have  been  suggested.82  There  is,  of  course,  danger  of  over- 
doing and  great  care  must  be  exercised  in  the  selection  and  in  the 
preparation  of  such  problems.  They  must  not  be  "fake"  engineer- 
ing problems,  nor  too  technical  nor  contain  too  many  elements.  To 
make  such  a  problem  usable  the  useful  parts  may  have  to  be  selected 


'  Chap.  Ill,  pp.  S9-6i.     Chap.  IV,  p.  83. 


PRESENT  NEEDS  AND  TENDENCIES.  117 

and  put  into  such  form  that  the  student  can  comprehend  it.  On 
the  other  hand  the  data  should  not  be  given  in  such  a  manner  as  to 
make  the  solution  a  mere  matter  of  substitution  in  a  formula. 

c)   Problems  to  Develop  Mathematical  Thought. 

The  student  who  can  not  think  mathematically  has  at  best  a 
superficial  knowledge  of  the  mathematical  language  and  consequent- 
ly is  unable  to  apply  it.  "Eternal  vigilance"  on  the  part  of  the  in- 
structor is  necessary  to  see  that  his  students  are  getting  a  grasp  of 
this  one  essential  and  are  really  thinking  the  problems  out  and  not 
using  a  "sleight-of-hand"  trick  for  doing  each  set  of  problems.  A 
good  illustration  along  this  line  is  furnished  in  the  study  of  asymp- 
totes. The  student  should  not  be  permitted  to  write  down  the  equa- 
tion of  the  asymptotes  of  4y2  —  Qx2  =  36  as  2y  ±  3x  =  o,  off-hand 
unless  he  can  find  the  values  of  M  andK which willmakey=Mx — K 
an  asymptote.  To  insure  that  he  really  does  this,  he  can  be 
required  to  find  the  asymptotes  of  4x2  —  6xy  -f-  4x  +  6y  —  y2  =  10, 
of  an  ellipse  or  of  some  curve  as  y  (2a  —  x)  =4ax.  Avoid  such 
text-books  as  differentiate  the  classes  of  problems  too  highly,  and 
such  as  introduce  each  set  by  some  one  worked  out  in  detail,  leaving 
nothing  for  the  student  but  the  substitution  of  numerical  values. 
When  a  problem  has  been  solved  vary  the  conditions  just  a  trifle 
and  test  the  class  on  the  new  aspect. 

6)  PEDAGOGIC  CONSIDERATIONS. 

The  following  questions  on  the  pedagogy  of  the  subject  are  of 
a  general  nature,  not  connected  with  any  particular  topics  or  princi- 
ples and  hence  will  here  be  discussed  separately. 

(*)  Modes  of  Instruction. 

It  is  not  supposed  that  any  instructor  will  hold  to  any  one 
mode  to  the  exclusion  of  all  others,  nor  to  any  fixed  combination 
at  all  times.  Some  combination  of  the  previously  mentioned  modes 
is,  however,  generally  followed  by  each  instructor.33  But  even 
though  an  instructor  may  have  been  very  successful  with  the  use  of 
one  mode  still  that  might  not  be  the  one  with  which  he  could  be  the 

w  Chap.  IV,  pp.  88-90. 


n8  MATHEMATICS  FOR  ENGINEERS. 

most  successful.  Nor  does  it  seem  likely  that  the  instructor  will  in- 
tuitively select  the  mode  best  suited  to  each  case.  The  selection 
consequently  deserves  careful  study. 

(«)  Suggested  Combinations  on  Modes. 

The  following  combinations  of  modes  are  recommended  as  the 
most  useful  :34 

A)  Assignment  of  work  with  or  without  explanations  and 
the  laboratory  mode. 

B)  Heuristic  and  laboratory  modes. 

C)  A  modification  of  the  heuristic  mode  applied  to  the 
class  as  a  whole  during  the  first  half  of  the  period,  the  second 
half  being  devoted  to  the  individual  students  while  working 
problems.     This  is  in  the  main  as  follows:     the  work  for  the 
following  day  is  assigned  first,  avoiding  hurry  at  the  end  of  the 
period  and  incidentally  giving  an  incentive  for  prompt  attend- 
ance.   The  theoretical  matter  for  the  day  is  then  taken  up  in 
the  form  of  an  oral  quiz,  in  connection  with  which  short  illus- 
trative problems  are  solved.    This  is  intended  to  clear  up  points 
of  difficulty,  to  fix  important  matters  and  to  insure  preparation 
from  day  to  day.     Whenever  possible  the  fundamental  prin- 
ciples of  the  next  assignment  are  taken  up  in  connection  with 
the  quiz  on  the  lesson  of  the  day  and  are  worked  out  by  the 
class  with  slight  suggestions  from  the  instructor.    Should  it  be 
impossible  to  connect  the  future  work  with  that  of  the  day,  the 
class  is  reviewed  on  past  work  with  which  it  can  be  connected. 
About  half  of  the  hour  is  consumed  in  this  manner.     The  re- 
mainder of  the  time  is  given  up  to  individual  work  by  the  stu- 
dents on  problems  either  at  the  boards  or  at  their  seats,  pre- 
ceded when  necessary  by  a  short  general  discussion  on  particu- 
lar difficulties  in  the  day's  problems.    During  this  work  on  prob- 
lems the  instructor  "visits"  each  student,  quizzing  him  and  giv- 
ing him   suggestions.     Individual   teaching  is  approached   as 
much  as  the  size  of  our  present  day  classes  will  permit.    As  the 
instructor  is  dealing  with  the  individual  he  has  even  more  free- 
dom to  select  the  mode  "best  suited  to  the  case"  than  if  he  had 

M  Chap.  IV,  p.  88. 


PRESENT  NEEDS  AND  TENDENCIES.  119 

to  consider  the  class  as  a  whole,  but  is,  of  course,  more  limited 
as  to  time.  One  of  the  real  dangers  is  the  temptation  to  slight 
the  good;  students  in  favor  of  the  mediocre,  or  vice  versa.  A 
college  class  of  from  twenty  to  twenty-five  and  a  high  school 
class  of  from  twenty-five  to  thirty  can  be  handled  successfully 
in  this  way. 

(MI)  Special  Texts  in  Mathematics. 

The  texts  used  in  mathematics  are  of  various  forms  and  de- 
grees of  specialization.  A  first  class  consists  of  those  which  treat 
each  topic  separately  and  with  little  connection  with  the  succeeding 
material  or  future  application,  but  lay  special  stress  upon  complete- 
ness of  the  proofs  and  logical  sequence.  As  a  rule  the  problems  are 
abstract  and  play  a  minor  part.  A  second  class  is  not  nearly  so  pro- 
fuse in  proofs  but  still  preserves  the  sharply  separated  topics.  The 
problems  are  not  so  abstract  but  often  impossible  or  too  hypothetical. 
This  is  what  may  be  termed  the  usual  old  standard  form  of  texts. 
The  third  class  confines  itself  to  definite  topics,  but  not  so  fully  as 
the  two  previous  classes,  and  these  topics  are  selected  with  a  view 
to  their  future  use  as  the  primary  object.  Points  of  special  interest 
to  engineers  are  brought  out,  as  use  of  the  slide  rule,  degree  of 
accuracy  in  computations,  etc.  For  the  most  part  the  problems  are 
simple,  real  and  sensibly  shorn  of  the  fantastic.  Those  of  the  fourth 
and  last  class  are  constructed  along  quite  radical  lines.  They  dis- 
regard the  divisions  of  algebra,  analytic  geometry  and  the  calculus 
and  form  a  homogeneous  whole  of  the  three  for  use  over  a  period 
of  from  one  to  two  years.  Often  these  are  written  for  the  needs  of 
the  students  at  some  particular  institution  or  of  a  particular  man, 
and  as  such  may  not  be  generally  useful. 

There  is  quite  a  division  7of  opinion  as  to  the  advisability  of 
using  texts  of  the  fourth  class.  In  several  instances  where  highly 
modified  courses  are  given,  texts  of  the  second  or  third  class,  sup- 
plemented by  problems  and  additional  material  of  a  more  decided 
engineering  nature,  are  used. 

The  early  introduction  of  the  ideas  of  the  calculus  already  dis- 
cussed (p.  in)  is  quite  feasible  under  such  a  plan.  The  methods 
of  the  calculus  that  are  used  would  be  developed  in  the  class  by  the 


120  MATHEMATICS  FOR  ENGINEERS. 

teacher  as  needed,  and  it  is  quite  possible  that  the  student  would 
thus  appreciate  these  new  appliances  better  than  if  he  were  intro- 
duced to  them  in  a  more  formal  way. 

(iV)   The  Profession  of  Teaching. 

The  seemingly  prevalent  idea  that  no  attention  need  be  given 
the  question  of  instruction  must  be  abandoned  before  results  of  the 
desired  excellence  are  obtained.33  Surely  the  history  of  mathemati- 
cal teaching  in  this  country,  and  especially  in  technical  work,  does 
not  show  it  to  be  so  simple  that  the  unprepared  beginner  can  do  the 
work  properly.  It  is  gratifying  to  note  that  attention  is  being  di- 
rected to  this  most  vital  question  and  it  is  hoped  it  will  soon  receive 
attention  commensurate  with  its  importance.86  The  reason  for  the 
present  excellent  instruction  in  Latin  is  the  fact  that  the  teachers  of 
Latin  are  trained  to  teach;  that  the  pedagogy  of  the  subject  is  one 
of  the  main  things  dwelt  upon;  that  the  Latin  departments  of  our 
advanced  institutions  give  "teachers'  courses"  not  only  for  pros- 
pective instructors  of  secondary  schools  but  also  for  those  of  colle- 
giate grade.87  One  of  the  absolute  requisites  for  good*  teaching  of 
mathematics  at  engineering  colleges  is  the  proper  preparation  of 
instructors  as  teachers  and  the  placing  of  mathematical  instruction 
on  the  plane  of  a  profession.  This  applies  especially  to  teachers  of 
freshmen  who  are  often  without  wide  experience,  while  the  math- 
ematics given  in  the  freshman  year  is  largely  the  foundation  of  the 
whole  professional  course. 

«  Chap.  IV,  p.  89. 

••  In  a  recent  address  Prof.  Alexander  Smith,  of  Columbia  University,  said  that 
life  is  a  series  of  lessons  in  "problem  solving"  and  that  no  subject  was  fit  for  the  college 
curriculum  which  could  not  be  reduced  to  that  basis.  Continuing  he  said:  "If  the 
American  College  is  to  be  rehabilitated  along  these  lines,  namely,  those  of  teaching 
problem  solving  and  giving  the  work  of  professional  standards,  or  along  similar  lines, 
two  important  improvements  in  the  present  situation  are  required.  To  give  all  subjects 
the  kind  of  instruction  indicated  will  requre  skillful  teachers,  and  it  is  generally  admit- 
ted that  the  teaching  of  undergraduates  is  at  present  less  satisfactory  than  is  that  of 
the  pupils  in  the  grades  and  high  school  on  the  one  hand,  or  of  the  student  of  the  grad- 
uate school  on  the  other.  The  second  need  is  that  of  more  efficient  methods  of  teaching, 
particularly  in  non-linguistic  subjects."  Alexander  Smith,  Rehabilitation  of  the  Ameri- 
can College,  Address  before  the  Educational  Sec.  Am.  Chem.  Soc.,  July  1909.  Science 
vol.  30,  p.  457.  See  also  E.  H.  Moore,  Presidential  Address  on  "The  Foundation  of 
Mathematics."  Bull.  Am.  Math.  Soc.  1903.  P-  4<>*- 

"  E.  H.  Moore,  Presidential  Address,  1.  c. 

John  Perry,  Preliminary  Education  of  Engineers,  Address  before  Engineering  Section 
Brit.  Soc.  for  Prom,  of  Sci.,  School  Science  and  Mathematics,  vol.  II,  p.  264.  1902. 


PRESENT  NEEDS  AND  TENDENCIES.  ISI 

V.     SUMMARY. 

The  present  chapter  has  been  reserved  for  the  discussion  of  the 
various  questions  which  have  arisen  in  connection  with  the  data 
previously  recorded.  For  good  work  during  the  freshman  year  it 
is  necessary  that  the  students  come  well  prepared  in  the  secondary 
work;  especially  in  algebra.  Two  plans  have  been  proposed  to 
insure  such  a  preparation.  The  first  is  to  require  entrance  examin- 
ations in  mathematics  of  all  students.  The  second  is  to  admit  with- 
out examination  in  mathematics  only  students  coming  from  high 
schools  which  have  special  teachers  of  mathematics  and  which  give 
a  half  year's  work  in  algebra  the  fourth  year.  The  following  is 
a  suggested  curriculum:  trigonometry  for  two  or  three  hours  per 
week  during  the  first  half  of  the  year,  and  a  course  in  algebra  and 
analytic  geometry  which  begins  with  a  review  of  preparatory  alge- 
bra for  the  first  two  or  three  weeks,  and  is  taken  two  or  three  hours 
per  week  the  first  half  year  and  for  five  hours  per  week  the  second 
half  year.  Analytic  geometry  is  to  be  the  basis  of  this  course; 
topics  of  algebra  being  taken  up  as  needed  or  as  they  arise  in  con- 
nection with  the  other  work  of  the  course.  The  object  of  analytic 
geometry  is  the  acquisition  of  a  new  mathematical  language  and  not 
information  regarding  any  particular  set  of  curves.  The  elements  of 
the  calculus  are  brought  into  use  in  the  solution  of  the  tangent  prob- 
lems and  in  the  finding  of  maxima  and  minima  of  functions.  As 
much  mathematics  is  to  be  taught  the  students  in  the  first  year  as  can 
be  included  in  the  curriculum  without  overcrowding,  and  they  are 
thus  to  be  given,  at  the  earliest  opportunity,  the  most  powerful 
weapons  for  attacking  the  problems  they  may  encounter  later.  The 
mathematical  curriculum  is  to  embody  such  principles  as  the  instruc- 
tors of  the  technical  subjects  consider  necessary  for  the  future  work, 
.and  is  to  be  taught  by  men  qualified  as  instructors  of  mathematics. 


CHAPTER  VI. 

% 

CONCLUSION. 

I.  CONTENTS  OF  CHAPTER. 

The  present  chapter  restates  briefly  the  most  salient  results  of 
the  investigation  recorded  in  what  precedes,  and  presents  the  con- 
clusions drawn  therefrom. 

II.    ORIGIN  AND  GROWTH  OF  THE  ENGINEERING 

COLLEGE. 

All  engineering  education  in  the  United  States  has  been  the 
result  of  a  demand  created  by  economic  conditions.  When  the  ap- 
prenticeship system  proved  inadequate  to  produce  sufficiently  trained 
engineers,  graduates  from  the  Military  Academy  were  drawn  into 
professional  work  along  civil  lines.  Stephen  Van  Rensselaer,  one 
of  the  early  promoters  of  American  industrial  enterprise,  gave  sub- 
stantial and  timely  aid  to  the  cause  of  industrial  education  by  the 
founding,  in  1824,  of  the  Rensselaer  School.  This  school  was  the 
pioneer  institution  of  its  kind  and  in  1835  offered  the  first  course  in 
civil  engineering  in  -the  United  States.  Soon  afterwards  several 
literary  institutions  organized  courses  in  civil  engineering  and  in- 
dustrial chemistry,  which  at  first  were  mostly  modifications  of  the 
last  year  or  two  of  the  regular  literary  courses.  By  the  middle  of 
the  ninetneenth  century  industries  had  arisen  that  required  men 
with  technical  education  in  mechanical  as  well  as  in  civil  engineer- 
ing. This  condition  prompted  Congress  to  pass  the  Land  Grant 
Act  of  1862  as  an  aid  to  mechanical  colleges.  Since  that  time  vari- 
ous courses  have  been  added  as  occasion  demanded,  such  as  Electri- 
cal Engineering,  Sanitary  Engineering,  etc.  As  a  member  of  our 
educational  system,  the  engineering  college  is  only  fifty  years  old.. 


CONCLUSIONS.  123 

III.    PROGRESS  OF  WORK  IN  MATHEMATICS  AND 
PRESENT  NEEDS. 

In  our  survey  of  the  progress  made  in  the  mathematical  work 
of  the  engineering  colleges  up  to  the  present  time  and  in  a  discussion 
of  future  improvements  we  have  considered  three  phases:  The 
preparation,  the  collegiate  curriculum,  and  the  problem  of  teaching 
the  courses  demanded. 

i)  ENTRANCE  REQUIREMENTS. — (i)  Their  Scope. 

During  the  formative  period,  or  until  the  coming  of  the  Land 
Grant  Colleges  in  the  early  sixties,  all  the  engineering  courses  of- 
fered, with  the  exception  of  those  at  the  Rensselaer  School,  were 
modifications  of  the  last  year  or  two  of  existing  literary  courses. 
Accordingly  two  or  three  years  of  literary  collegiate  work  were  re- 
quired for  admission  to  these  technical  courses.  As  more  and  more 
work  was  added  to  the  technical  courses  the  amount  of  the  literary 
collegiate  work  required  for  admission  to  the  technical  work  had 
to  be  reduced  until  when  the  complete  four  years'  technical  course 
was  offered  the  only  work  common  to  the  two  courses  was  that  for 
entrance.  The  door  was  then  opened  for  differentiating  the  entrance 
requirements,  and  this  has  also  been  done  to  some  extent.  In  math- 
ematics they  have  been  raised  for  the  technical  courses  from  arith- 
metic (prior  to  1850)  to  the  present  average  of  one  and  one-half 
years  of  algebra,  one  year  of  plane  geometry  and  one- half  year  of 
solid  geometry.  Trigonometry  is  required  by  i&%  of  the  institu- 
tions.1 

(n)  Present  Needs. 

The  above  entrance  requirements  are  at  the  present  adequate 
as  to  extent.  The  need  is  rather  for  more  concrete  presentations  of 
the  subjects  and  with  a  view  to  their  applications,  and  for  more 
emphasis  upon  the  principles  that  will  be  met  in  the  collegiate  work, 
especially  the  principles  concerning  the  handling  of  algebraic  ex- 
pressions and  their  application  to  problems,  in  both  of  which  many 
deficiencies  are  found.  This  demand  that  preparatory  algebra  be 

1  Chap.  II,  p.  43. 


124  MATHEMATICS  FOR  ENGINEERS. 

taught  concretely  and  for  its  applied  value,  that  the  various  topics 
be  treated  as  parts  of  a  homogeneous  whole  centered  about  the 
equation,  that  fundamental  principles  be  made  to  stand  out  boldly 
and  clear  from  the  working  machinery  involved,  and  that  the  work 
throughout  be  emphasized  by  numerous  fairly  simple  problems. 
Such  teaching  of  mathematics  for  future  utility  is  conceded  also 
to  be  the  most  generally  valuable  from  the  educational  point  of  view. 

(m)  Remedy. 

How  to  secure  the  good  preparation  in  entrance  mathematics 
that  is  so  vital  to  successful  work  in  mathematics  during  the  fresh- 
man year,  is  one  of  the  most  serious  questions  confronting  the  en- 
gineering colleges  of  today.  Unless  the  high  school  teachers  have 
been  especially  trained  for  the  teaching  of  mathematics,  satisfactory 
preparation  of  students  is  impossible.  It  is  the  duty  of  the  engineer- 
ing colleges  vigorously  to  advocate  such  training  and  to  require 
en<trance  examinations  in  mathematics  of  the  graduates  of  all  high 
schools  not  having  a  special  teacher  of  mathematics.  In  addition 
to  this,  only  such  high  schools  should  be  accredited  as  give  a  half 
year's  review  of  algebra  in  the  fourth  year.  As  a  final  means  of 
securing  good  preparation  for  the  work  of  the  freshman  year  two 
or  three  weeks'  review  at  the  beginning  of  the  year  is  recommended, 
and  students  then  found  especially  deficient  should  be  required  to 
take  additional  preparatory  work. 

(iv)   Usefulness  of  Variation  in  Entrance  Requirements. 

The  great  difference  in  entrance  requirements2  found 
among  the  various  engineering  colleges  seems,  at  first  glance,  a 
condition  to  be  deplored.  Yet  if  the  high  schools  take  advantage 
of  this  divergency  they  will  find  it  of  inestimable  help  in  aiding 
those  of  their  students  who  intend  taking  an  engineering  course,  to 
select  an  institution  commensurate  with  their  capacity.  Strangely 
enough,  the  great  number  of  failures  in  mathematics  on  the  part  of 
freshmen  students  makes  it  appear  as  though  little  cognizance  were 
taken  of  this  divergency.2  The  lack  of  uniformity  in  entrance 
requirements  may  also  serve  as  the  entering  wedge  for  the  organi- 
zation of  some  system  of  grading  our  engineering  colleges.  When 

*  Chap.  II,  p  43. 


CONCLUSIONS.  125 

this  is  done  technical  education  will  have  been  placed  upon  a  firmer 
foundation  and  much  of  the  present  waste  of  time  and  energy  will 
be  saved,  especially  in  the  work  in  mathematics. 

2)  COLLEGIATE  CURRICULUM. — (i)  Progress. 

Until  1850  the  collegiate  courses  included  all  of  algebra,  to- 
gether with  plane  and  solid  geometry,  trigonometry  and  a  little  anal- 
ytic geometry  given  in  the  latter  part  of  the  second  year.  After  that 
time  some  algebra  was  required  for  entrance  and  the  collegiate  work 
was  increased  correspondingly ;  calculus  was  first  added  in  1855, 
but  came  late  in  the  course  and  was  little  used  in  the  technical  work. 
Since  then  the  time  devoted  to  analytic  geometry  and  the  calculus 
has  been  materially  increased,  and  these  subjects  have  been  placed 
early  in  the  curriculum  so  as  to  make  them  available  for  the  technical 
work.  At  present  from  seventy  to  one  hundred  and  twenty  class  hours 
are  devoted  to  analytic  geometry,  which  is  completed  during  the 
first  year  or  in  the  early  part  of  the  second  year,  and  from  one  hun- 
dred to  two  hundred  class  hours  are  given  to  the  calculus  in  the 
second  year.  It  is  only  within  the  last  fifteen  years,  however,  that 
attention  has  been  given  to  the  introduction  of  work  especially 
planned  for  engineering  students,  and  these  modified  curricula  are 
as  yet  only  in  the  formative  state.3  Some  of  the  more  noteworthy 
features  are:  i)  the  treatment  of  the  subjects  from  the  standpoint 
of  application  rather  than  that  of  philosophy,  that  is,  less  attention  is 
paid  to  the  rigorous  proofs  of  theorems  than  to  their  application ; 
2)  less  insistance  upon  sharp  lines  of  demarkation  between  the  sub- 
jects;  3)  the  introduction  of  laboratory  courses;  4)  the  correlation 
of  the  work  in  algebra  and  analytic  geometry,  without  special  atten- 
tion to  the  conies;  5)  the  intrbduction  of  the  calculus  into  the  work 
of  the  freshman  year,  especially  in  connection  with  the  solution  of 
the  tangent  problem.4  The  causes  for  the  adoption  of  this  modified 
work  may  be  summarized  as  a  desire  to  lessen  "the  gap  between 
theory  and  practice."5 

»  Chap.  III. 

4  Chap.  Ill,  pp.  52-62. 

«  Chap.  Ill,  p.  66. 


126  MATHEMATICS  FOR  ENGINEERS. 

(it)  Suggested  Improvements. 

Ever  since  the  founding  of  the  Rensselaer  School  the  object 
of  the  engineering  colleges  has  been  the  production  of  technically 
trained  men.  The  principles  necessary  to  such  a  training  must, 
therefore,  control  the  formation  of  the  curriculum.  To  no  subject 
is  this  more  applicable  than  to  mathematics,  so  fundamental  for  the 
more  technical  work.  Future  use,  not  culture  value,  must  be  the 
criterion  according  to  which  mathematical  topics  are  admitted  or 
rejected.  It  hence  becomes  necessary  for  the  instructors  of  the 
professional  studies  to  cooperate  with  those  of  the  mathematical 
department  by  designating  what  mathematical  principles  are  needed 
in  the  professional  courses.  All  mathematical  topics  should  be 
treated  in  as  close  proximity  as  possible  to  their  application.  When 
the  usual  division  into  subjects  is  not  adhered  to  principles  which 
are  allied  to  each  other  will  naturally  be  connected,  and  thus  the 
value  of  each  will  be  enhanced.  It  is  a  decided  advantage  to  equip 
the  student  with  the  most  powerful  instrument  applicable  to  a  prob- 
lem or  set  of  problems  the  first  time  such  problems  arise,  for  ex- 
ample, the  use  of  derivatives  in  the  solution  of  tangent  problems. 
The  most  efficient  means  will  thus  become  the  natural  one  for  the 
student  to  use  and  he  will  become  more  proficient  in  handling  them 
thru  prolonged  application.  The  mathematical  curriculum  for  the 
freshman  year  should  contain  as  much  usable  mathematics  as  pos- 
sible without  overcrowding. 

3)  INSTRUCTION — (i)  Progress  and  Present  Needs. 

The  men  who  furnished  the  information  under  the  apprentice- 
ship system  could  in  no  wise  be  called  mathematicians  or  teachers. 
From  the  coming  of  the  engineering  colleges  there  has  been  con- 
tinual advancement  in  mathematical  knowledge  among  the  instruc- 
tors until  few  deficiencies  are  at  present  found  on  that  side.  Some 
attention  is  also  being  given  to  the  question  of  preparing  men  as 
teachers,6  but  on  the  whole  little  has  been  done  in  this  direction  as  is 
shown  by  the  lack  of  interest  in  the  technical  questions  of  instruc- 

*  Chap.  V,  p.  1 20. 


CONCLUSIONS. 


127 


tion.7  Until  due  attention  is  given  to  this  most  vital  phase  of  the 
problem  and  the  work  of  mathematical  instruction  is  raised  to  the 
rank  of  a  profession,  the  desired  results  can  not  be  hoped  for. 

(M)  Special  Phases  to  be  Noted. 

The  following  are  some  of  the  salient  points  to  be  considered 
relative  to  the  teaching  of  mathematics  to  freshmen  students  of 
engineering. 

a)  General  Aims. 

The  general  aim  of  the  instruction  in  mathematics  given  to 
freshman  students  should  be  to  bring  out  its  utilitarian  value.  To 
accomplish  this  end  it  is  necessary  to  make  the  relations  between 
the  principles  and  their  application  the  primary  object,  relegating 
exhaustive  proofs  to  the  background.  In  thus  learning  to  apply 
the  principles  the  student  will  also  secure  a  clearer  understanding  of 
the  principles  themselves  than  he  could  possibly  attain  thru  any 
other  means.  Such  a  utilitarian  mathematics  fully  understood  by 
the  student  and  applied  by  him  in  a  logical  and  intelligent  manner 
is  also  of  the  highest  educational  value.8  No  study  should  be  in- 
troduced for  its  value  as  a  mental  gymnastic. 

b)  Fundamental  Principles. 

For  the  engineering  student  even  more  than  for  the  literary 
student  attention  to  fundamental  principles  is  of  the  utmost  im- 
portance. These  principles  should  therefore  first  be  studied  thoroly 
and  illustrated  by  a  great  number  of  slightly  varied  problems ;  and 
secondly,  the  reduction  of  each  problem  to  these  fundamentals 
should  thereafter  be  insisted  upon  at  all  times.  Such  a  thoro 
grounding  in  fundamentals  may  necessitate  the  elimination  of  some 
of  the  "advanced  work,"  but  without  the  ground  work  this  would 
be  largely  superficial,  while  the  student  thoroly  equipped  with  the 
basic  principles  will  be  able  to  do  the  more  involved  work  as  it 
arises. 

c)  Problems. 

The  value  of  any  educational  work  is  directly  proportional  to 
the  interest  taken  by  the  students.  In  mathematics  this  interest  is 

T  Chap.  IV,  p.  89. 

8  Such  a  rational  study  has  been  ably  described  on  page  95. 


128  MATHEMATICS  FOR  ENGINEERS. 

largely  dependant  upon  the  problems  proposed  for  solution.  As 
the  work  of  the  freshmen  engineers  is  to  be  given  mainly  from 
the  standpoint  of  application  it  is  readily  seen  what  an  important 
part  problems  play  and  what  great  care  is  needed  in  their  selection. 
The  data  selected  for  the  problems  should  be  as  real  as  possible  and 
entirely  shorn  of  the  fantastic.  The  use  of  problems  from  engineer- 
ing data  is  of  value  not  only  in  giving  the  students  some  little 
insight  into  the  nature  of  their  future  work  but  especially  in 
holding  their  interest.  Still,  before  being  used  in  the  mathematical 
work  such  data  must  be  carefully  edited  so  as  to  exemplify  the 
principles  which  it  is  desired  to  illustrate,  without  requiring  the 
student  to  become  acquainted  with  the  technical  part  of  the  engi- 
neering questions  involved.  Too  long  and  too  complicated  prob- 
lems, on  the  one  hand,  and  those  requiring  mere  substitution  in  a 
formula  on  the  other,  must  be  guarded  against,  and  "fake"  engineer- 
ing problems  must  never  be  used.  The  instructor  must  also  be 
careful  not  to  overdo  matters  and  must  take  especial  pains  to  avoid 
the  duplication  of  the  work  of  the  professional  departments.  Prob- 
lems should  never  be  selected  for  their  symetry  of  form ;  such 
problems  are  rarely  met  with  afterwards  and  should  rather  be 
avoided,  perfectly  general  but  simple  data  being  used  instead. 

d)  Computation. 

Accuracy  in  computation  is  a  vital  question  with  the  profes- 
sional engineer,9  and  the  instructors  of  technical  subjects  can  rightly 
require  some  help  from  the  instructors  of  mathematics  along  this 
line.  This  can  be  given  either  by  special  courses  in  computation, 
such  as  have  been  inaugurated  by  some  institutions,10  or  what  ap- 
pears on  the  whole  more  satisfactory,  by  carrying  along  some  work 
in  computation  thruout  the  whole  mathematical  curriculum.  To 
carry  out  this  plan  the  following  few  points  should  be  observed  by 
the  instructor: 

/)   Never  let  the  value  of  numerical  accuracy  be  slighted. 
2}  Drill  the  students  to  present  final  results  in  the  form  best 
suited  for  computation;  as  tf  V6  in  the  place  of 

•  Chap.  IV,  p.  85. 

"  Chap.  Ill,  pp.  62-63. 


CONCLUSIONS.  129 

5)  Introduce  the  use  of  tables  whenever  they  materially  shorten 
the  work  and  their  use  can  be  thoroly  understood  by  the 
students. 

4)  Permit  the  use  of  formulas  only  after  the  student  is  thoroly 
acquainted  with  the  quantities  involved   and  the  relations 
expressed  between  them.     No  recognition  is  to  be  given 
solutions  based   upon   formulas  except  where  perfect  ac- 
curacy is  obtained.     Whenever  a  common  sense  application 
of  arithmetic  or  the  elements  of  algebra  and  geometry  will 
suffice  for  the  solution  of  a  problem,  the  use  of  a  formula 
should  be  discouraged.11 

5)  Teach  what  is  meant  by  the  degree  of  accuracy  needed  and 
what  degree  is  possible  in  any  given  problem.12 

4)  Too  MUCH  REQUIRED  OP  THE  MATHEMATICAL  DEPARTMENT. 

According  to  the  statistics  previously  recorded13  the  language 
departments  recieve  only  slightly  less  time  than  the  mathematical 
departments  while  the  semi-professional  departments  receive  some- 
what more  than  it  does.  As  mathematics  is  considered  to  be  the 
foundation  of  engineering  science,  is  it  not  fair  to  inquire  in  the 
light  of  these  figures  if  too  much  is  not  asked  of  the  instructors  of 
mathematics?  As  none  of  the  other  departments  seem  able  to  re- 
linquish any  of  their  portion  of  time,  then'  from  all  experiences 
noted,  two  things  are  necessary:  i)  the  mathematical  courses 
must  be  made  "more  intensive  and  less  extensive"  and  2)  the 
mathematical  departments  must  have  the  cooperation  of  the  pro- 
fessional departments  in  order  to  make  the  courses  in  mathematics 
fit  the  demands  made  upon  them. 

III.     MATHEMATICS    FOR   FRESHMEN    STUDENTS    OF 

ENGINEERING. 

The  mathematical  courses  for  freshmen  students  of  engineer- 
ing should  be  composed  of  the  topics  and  principles  which  are  con- 
sidered by  the  instructors  of  technical  subjects  to  be  prerequisite 

n  For  confirmation  and  illustrations  see  p.  92. 
u  Chap.  V,  p.   114. 
13  Chap.  II,  p.  49. 


MATHEMATICS  FOR  ENGINEERS. 

for  such  subjects  and  this  material  together  with  what  it  presupposes 
should  be  formulated  into  a  compact  teachable  whole  by  the  teachers 
of  mathematics.  The  curriculum  is  to  contain  as  much  usable 
mathematics  as  possible;  only  the  most  powerful  methods  of  at- 
tack, those  which  will  be  required  for  use  in  the  future,  being 
taken  up.  Basic  principles  involving  the  fundamental  conception  of 
the  subjects  are  to  receive  chief  attention,  but  little  time  being 
given  to  so  called  "advanced  work."  Each  principle  studied,  should 
be  amply  illustrated  by  simple  problems  based,  as  far  as  possible, 
upon  real  data.  Intelligent  and  accurate  application  of  the  principles 
rather  than  their  rigorous  proofs  should  characterize  the  work  thru- 
out.  The  need  of  accuracy  in  computed  results  and  the  degree  of 
accuracy  attainable  should  be  emphasized,  and  the  placing  of  re- 
sults in  the  best  form  for  computation  should  always  be  insisted 
upon.  Instructors  trained  as  teachers  as  well  as  thoroly  versed 
in  the  subject  matter  to  be  taught  are  requisite  for  successful  work. 
The  mathematics  for  the  freshmen  students  of  engineering  should 
be  a  "vitalized  mathematics"  dealing  with  as  real  and  interesting 
data  as  possible;  it  should  be  a  utilitarian  mathematics  taught 
rationally  and  intelligently  so  as  to  give  the  students  power  in  using 
the  mathematical  type  of  thot  as  well  as  training  in  its  technical 
application. 


BIBLIOGRAPHY. 

Alderson,  V.  C.    German  Technical  Schools.    Chicago,  1901. 

Alderson,  V.  C.     The  Progress  and  Influence  of  Technical  Education, 
Proceedings   Society  for  the   Promotion  of  Engineering  Education,  XIII : 
126-45.     I9°5- 

Armstrong,  H.  E.  The  Teaching  of  Scientific  Methods.  London  and 
New  York,  1903. 

Baker,  I.  O.  Engineering  Education  in  the  United  States  at  the  End  of 
the  Century.  Proceedings  Society  for  the  Promotion  of  Engineering  Edu- 
cation, VIII:  11-27.  1900. 

Benjamin,  Park.    United  States  Naval  Academy.     New  York,  1900. 

Bohannan,  R.  D.  A  Calculation  Blunder  Common  to  Many  Texts  on 
Trigonometry  used  in  Engineering  Courses.  Proceedings  Society  for  the 
Promotion  of  Engineering  Education,  XV:  655-61.  1907. 

Bohannan,  R.  D.  A  Neglected  Opportunity  to  Teach  Consistant 
Measurement  in  Teaching  Trigonometry.  Proceedings  Society  for  the  Pro- 
motion, of  Engineering  Education,  XV;  662-67.  1907. 

Burr,  W.  H.  The  Ideal  Engineering  Education.  Proceedings  Society 
for  the  Promotion  of  Engineering  Education,  1 :  1-49.  1893. 

Cajori,  F.  Some  Hints  on  Teaching  Mathematics  to  Engineering  Students. 
Proceedings  Society  for  the  Promotion  of  Engineering  Education,  XIII :  26- 
33-  1905- 

Cajori,  F.  The  Teaching  and  History  of  Mathematics  in  the  United 
States.  Government  Printing  Office,  1890. 

Campbell,  D.  F.     Calculus  for  Engineers.     New  York  and  London,  1905. 

Clark,  I.  E.  Education  in  the  Industrial  and  Fine  Arts  in  the  United 
States,  IV:  838-48.  United  States  Bureau  of  Education,  1898. 

Comstock,  C.  E.  A  Few  Difficulties  in  the  Teaching  of  Mathematics. 
School  Science  and  Mathematics,  7 :  12-18.  1907. 

Comstock,  C.  E.  Unification  of  Secondary  Mathematics.  Proceedings 
National  Educational  Association,  pp.  519-21,  1904. 

Creighton,  W.  H.  P.  Methods  of  Teaching.  Proceedings  Society  for 
the  Promotion  of  Engineering  Education,  V:  101-16.  1897. 

Cullum,  Bvt.  Maj.-Gen.  G.  W.  Early  History  of  the  United  States 
Military  Academy.  Boston  and  New  York,  1891. 

Drown,  T.  F.  Technical  Training.  Journal  of  Franklin  Institute,  116: 
329-54.  1883. 

Duncan,  L.  Technical  Education — What  It  Should  be.  Engineering 
Magazine,  26:  161-68.  1903. 


132  MATHEMATICS  FOR  ENGINEERS. 

Eddy,  H.  T.  Engineering  Education.  Proceedings  Society  for  the  Pro- 
motion of  Engineering  Education,  V:  11-25.  1897. 

Emory,  F.  L.  Advanced  Algebra  in  Engineering  and  Other  Colleges. 
Proceedings  Society  for  the  Promotion  of  Engineering  Education,  VII : 
104-09.  1899. 

Fletcher,  R.  The  Present  Status  and  Tendencies  of  Engineering  Educa- 
tion. Proceedings  Society  for  the  Promotion  of  Engineering  Education, 
VIII:  181-90.  1900. 

Franklin,  W.  S.  and  Macnutt,  B.  Teaching  of  Elementary  Mechanics. 
Proceedings  Society  for  the  Promotion  of  Engineering  Education,  XV: 
316-34.  1907. 

Goldziher,  K.  Mathematical  Laboratories.  School  Science  and  Mathe- 
matics,'VIII :  753-57-  1908. 

Gould,  E.  S.  Engineering  Mathematics.  Cassiers  Magazine,  27:  227-32. 
1904- 

Groat,  B.  F.  Teaching  Calculus  to  Engineering  Students.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XII:  225-30.  1904. 

Hall  and  Frink.    Trigonometry.     New  York,  1909. 

Hathway,  A.  S.  Mathematics  for  Engineering  Students.  Bulletin 
American  Mathematical  Society,  VII :  266-71.  1901. 

Haynes,  A.  E.  What  Should  be  the  Characteristic  Features  of  the 
Teaching  of  a  Course  in  Mathematics  for  Engineering  Students?  Pro- 
ceedings Society  for  the  Promotion  of  Engineering  Education,  VIII:  308-14. 
1900. 

Haynes,  A.  E.  The  Justification  of  the  Use  of  the  Expression  "Engi- 
neering Mathematics."  Proceedings  Society  for  the  Promotion  of  Engineer- 
ing Education,  XIV:  127-38.  1906. 

Heaviside,  O.  Electromagnetic  Theory.  Introduction  in  vol.  I  and 
Mathematics  in  vol.  II.  London  1893-99. 

Hedrick,    E.    R.    Approximations    and    Approximation     Processes.     School 
Science  and  Mathematics,  VIII:  617-25  and  745-52.     1908. 

Hollis,  I.  N.  The  Naval  Academy  as  a  Technical  School.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XII :  159-92.  1904. 

Jackman,  W.  S.  Correlation  of  Mathematics.  Educational  Review,  25: 
249-64-  I903- 

Jackson,  D.  C.  Desirable  Products  from  the  Teacher  of  Mathematics — 
Point  of  View  of  an  Engineering  Teacher.  School  Science  and  Mathematics, 
V :  67-74-  I905- 

Jackson,  J.  P.  Methods  of  Study  for  Technical  Students.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XI :  101-16.  1903. 

Kent,  W.  Pedagogical  Methods  in  Engineering  Colleges.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XV:  90-98.  1907. 

Kenyon,  A.  M.  Teaching  Calculus  to  Engineering  Students.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XII :  220-25.  1904. 


BIBLIOGRAPHY.  133 

Low,  D.  A.  The  Teaching  of  Mathematics  in  the  United  Kingdom; 
(No.  8)  Geometry  for  Engineers.  London,  1911.  (Too  late  for  use  in 
above.) 

Lunn,  A.  C.  Outline  of  a  Coherent  Course  in  College  Algebra.  Ameri- 
can Mathematical  Monthly,  XII :  123-29.  1905. 

McNair,  P.  W.  The  Calculus  for  Engineering  Students.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  V :  139-49.  1897. 

Marx,  C.  D.  General  Education  for  Engineers.  Popular  Science 
Monthly,  67 :  442-46.  1905. 

Merriman,  Mansfield.  Teachers  and  Text-books  in  Mathematics  in 
Technical  Schools.  Proceedings  Society  for  the  Promotion  of  Engineering 
Education,  II:  95-103.  1894. 

Miller,  G.  A.  Reform  in  Mathematical  Instruction.  Science,  24:493-96. 
1906. 

Moore,  E.  H.  Qross-section  Paper  as  a  Mathematical  Instrument. 
School  Science  and  Mathematics,  VI :  429-50.  1906. 

Moore,  E.  H.  The  Foundations  of  Mathematics.  Presidential  Address 
at  Annual  Meeting  American  Mathematical  Society,  1902.  Bulletin  American 
Mathematical  Society,  IX:  402-24.  1903. 

Myers,  G.  W.  A  Class  of  Content  Problems  for  High  School  Algebra. 
School  Science  and  Mathematics,  VII :  19-33.  IQO7- 

Myers,  G.  W.  Laboratory  Method  in  Secondary  Schools.  School  Re- 
view, XI :  727-41.  1903. 

Newcomb,  S.  Teaching  of  Arithmetic.  Proceedings  National  Educa- 
tional Association,  pp.  86-102,  1906. 

Newcomb,  S.  Teaching  of  Mathematics.  Educational  Review,  IV : 
277-86  and  VI:  332-41.  1892  and  1893. 

Newson,  H.  B.    Graphic  Algebra.     Boston,  1905. 

Perry,  J.     Calculus  for  Engineers.     London,  1897. 

Perry,  J.  Electrical  Engineering  as  a  Trade  and  as  a  Science.  Nature, 
63:  41-47.  1900. 

Perry,  J.  Preliminary  Education  of  Engineers.  School  Science,  II: 
264-72.  1902. 

Perry,  J.     Reform  in  Mathematics.     London,  1902. 

Perry,  J.     Practical  Mathematics.     London,  1907. 

Preece,  Sir  W.  H.    Functions  of  the  Engineer.    Nature,  61  -.374-77.    1900. 

Prout,  H.  G.  The  Future  of.  Engineering.  Scientific  American  Supple- 
ment, 52:21451-52.  1901. 

Randall,  O.  E.  Descriptive  Geometry.  Proceedings  Society  for  Promo- 
tion of  Engineering  Education,  XV :  617-28.  1907. 

Rickets,  P.  C.  History  of  Rensselaer  Polytechnic  Institute  1824-94. 
New  York,  1895. 

Rietz,  H.  L.  Present  Situation  in  Regard  to  Teaching  Algebra  in  our 
High  Schools.  School  Science  and  Mathematics,  VIII:  496-502.  1908. 

Ripper,  W.  Technical  Education  in  Relation  to  Industrial  Progress. 
Nature,  61 :356-57.  1900. 


134  MATHEMATICS  FOR  ENGINEERS. 

Russell,  J.  H.     Systematic  Technical  Education.     London,  1865. 

Sanders,  A.     Elements  of  Plane  Geometry.     New  York,  1901. 

Saxelby,  F.  A  Course  in  Practical  Mathematics.  London  and  New 
York,  1905. 

Schneider,  H.  Cooperative  Course  in  Engineering  at  the  University  of 
Cincinnati.  Proceedings  Society  for  the  Promotion  of  Engineering  Educa- 
tion, XV :  291-406.  1907. 

Slichter,  C.  S.  Improvement  of  the  Freshman  Year  in  Mathematical 
Instruction  in  Technical  Schools.  Proceedings  Society  for  the  Improvement 
of  Engineering  Education,  XIV:  146-62.  1906. 

Smith,  D.  E.  The  Teaching  of  Elementary  Mathematics  New  York, 
1900. 

Smith,  R.  H.  Graphics.     London  and  New  York,  1889. 

Soley,  Rear-Admiral  J.  R.  Historical  Sketch  of  the  United  States  Naval 
Academy.  Government  Printing  Office,  1876. 

Spencer,  H.  What  Knowledge  is  Most  Worth.  Collected  Essays  on 
Education,  p.  84.  New  York,  1883. 

Stanwood,  J.  B.  Mathematics  in  Engineering  Colleges.  Cassiers  Maga- 
zine, 28:  148-50.  1904. 

Steinmitz,  C.  P.  Theoretical  Elements  of  Electrical  Engineering.  New 
York,  1909. 

Sutherland,  J.  C.  The  Engineering  Mind.  Popular  Science  Monthly, 
62:  254-56.  1903. 

Swain,  G.  F.  Some  Observations  Regarding  the  Value  of  Mathematics 
to  the  Civil  Engineer  and  on  the  Teaching  of  the  Subject  to  Civil  Engineers. 
Proceedings  International  Mathematical  Congress,  p.  361.  1909. 

Talbot,  A.  N.  Requirements  in  Mathematics  for  an  Engineering  Educa- 
tion. Proceedings  Society  for  the  Promotion  of  Engineering  Education,  I : 
50-62.  1893. 

Thurston,  R.  H.  On  the  Organization  of  Engineering  Courses,  and  On 
Entrance  Requirements  for  Professional  Schools.  Ithaca. 

Thurston,  R.  H.  Professional  and  Academic  Schools.  Educational  Re- 
view, 17:  16-36.  1899. 

Townsend,  E.  J.  Analysis  of  the  Failures  in  Freshman  Mathematics. 
School  Review,  X :  675-86.  1902. 

Trotter,  A.  P.  Mathematics  and  Engineering.  Engineering  Magazine, 
26:  266-68.  1902. 

Trotter,  A.  P.  Usefulness  of  Mathematics  from  the  Engineers'  Stand- 
point. Engineering,  76 :  358-60.  1903. 

Walker,  F  A.  Place  of  Technical  Schools  in  American  Education.  Edu- 
cational Review,  II :  209-19.  1891. 

Warren,  S.  E.  Coordination  of  Polytechnic  Schools.  Education,  24: 
232-44.  1903. 

Wertheimer,  J.  Higher  Technical  Education  in  Germany  and  Great 
Britain.  Nature,  68 :  274-76.  1903. 


BIBLIOGRAPHY.  135 

Wilson,  E.  B.  Some  Recent  Books  on  Mechanics.  Bulletin  American 
Mathematical  Society,  VIII :  403-09  and  IX  :  25-39.  1902. 

Woods  and  Bailey.  A  Course  in  Mathematics.  Boston  and  New  York. 
1907. 

Young,  J.  W.  A.  The  Teaching  of  Mathematics  in  the  Elementary  and 
Secondary  Schools.  New  York.  1907. 

Young,  J.  W.  A.  What  is  the  Laboratory  Method?  Mathematical 
Supplement  of  School  Science,  III :  50-56.  1903. 

N-E  Coast  Institute  of  Engineers  and  Ship-builders  (England).  Techni- 
cal Education  in  Germany.  Engineering  Magazine,  26:  999-1000.  1902. 

Notable  Addresses  in  England  and  Germany.  Problems  in  Technical 
Education.  Engineering  Magazine,  24:267-68.  1902. 

Report  of  Standing  Committee  on  Entrance  Requirements.  Proceedings 
Society  for  tfhe  Promotion  of  Engineering  Education,  IX:263-9i  and  X: 
195-206.  1901  and  1902. 

Report  of  Committee  on  Requirements  for  Graduation.  Proceedings 
Society  for  the  Promotion  of  Engineering  Education,  XII :  99-130.  1904 

Committee  No  IX,  International  Commission  on  the  Teaching  of  Mathe- 
matics in  the  Technological  Schools  of  Collegiate  Grade  in  the  United  States. 
(To  late  for  use  in  above.) 

VITA. 

Theodore  Lindquist  was  born  in  Andover,  111.,  where  he  received  most  of 
his  elementary  education.  He  was  graduated  from  the  Galesburg,  111.,  High 
School  in  1894,  received  the  degree  of  A.  B.  from  Lombard  College  (Gales- 
burg,  111.)  in  1897  and  the  degree  of  M.  S.  from  Northwestern  University 
(Evanston,  111.)  in  1899  after  one  year  of  residente  study  during  which  he 
attended  lectures  by  Professors  Crew,  Holgate  and  White.  Subsequently 
he  has  spent  six  quarters  (the  equivalent  of  two  academic  years)  at  The 
University  of  Chicago  working  in  the  Departments  of  Mathematics  and 
Physics  and  attending  lectures  by  Professors  Bliss,  Bolza,  Dickson,  Maschke, 
Michelson,  Millikan,  Moulton  and  Young.  He  feels  grateful  to  all  of  these 
instructors  for  the  help  and  encouragement  received  thruout  his  work,  and 
especially  to  Professor  Young  under  whose  direction  the  dissertation  was 
prepared;  he  also  thanks  those  who  so  kindly  contributed  to  the  data  en> 
ployed. 


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